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Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver, Chehrzad Shakiban

Chapter 8

Eigenvalues and Singular Values - all with Video Answers

Educators

Chapter Questions

07:59

Problem 1

Solve the following initial value problems:
(a) $\frac{d u}{d t}=5 u, u(0)=-3$,
(b) $\frac{d u}{d t}=2 u, u(1)=3$,
(c) $\frac{d u}{d t}=-3 u, u(-1)=1$.

Andrija Isakov
Andrija Isakov
Numerade Educator

Problem 1

Find the eigenvalues and eigenvectors of the following matrices:
(a) $\left(\begin{array}{rr}1 & -2 \\ -2 & 1\end{array}\right)$,
(b) $\left(\begin{array}{ll}1 & -\frac{2}{3} \\ \frac{1}{2} & -\frac{1}{6}\end{array}\right)$,
(c) $\left(\begin{array}{rr}3 & 1 \\ -1 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rr}1 & 2 \\ -1 & 1\end{array}\right)$
(e) $\left(\begin{array}{rrr}3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3\end{array}\right)$,
(f) $\left(\begin{array}{rrr}-1 & -1 & 4 \\ 1 & 3 & -2 \\ 1 & 1 & -1\end{array}\right)$,
(g) $\left(\begin{array}{rrr}1 & -3 & 11 \\ 2 & -6 & 16 \\ 1 & -3 & 7\end{array}\right)$,
(h) $\left(\begin{array}{rrr}2 & -1 & -1 \\ -2 & 1 & 1 \\ 1 & 0 & 1\end{array}\right)$,
(i) $\left(\begin{array}{rrr}-4 & -4 & 2 \\ 3 & 4 & -1 \\ -3 & -2 & 3\end{array}\right)$,
(j) $\left(\begin{array}{llll}3 & 4 & 0 & 0 \\ 4 & 3 & 0 & 0 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 4 & 5\end{array}\right)$,
(k) $\left(\begin{array}{rrrr}4 & 0 & 0 & 0 \\ 1 & 3 & 0 & 0 \\ -1 & 1 & 2 & 0 \\ 1 & -1 & 1 & 1\end{array}\right)$.

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Problem 1

Which of the following are complete eigenvalues for the indicated matrix? What is the dimension of the associated eigenspace?
(a) $3,\left(\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right)$;
(b) 2, $\left(\begin{array}{ll}2 & 1 \\ 0 & 2\end{array}\right)$;
(c) $1,\left(\begin{array}{rrr}0 & 0 & -1 \\ 1 & 1 & 0 \\ 1 & 0 & 0\end{array}\right)$
(d) $1,\left(\begin{array}{rrr}1 & 1 & -1 \\ 1 & 1 & 0 \\ 1 & 1 & 2\end{array}\right)$;
(e) $-1,\left(\begin{array}{rrr}-1 & -4 & -4 \\ 0 & -1 & 0 \\ 0 & 4 & 3\end{array}\right)$;
(f) $-\mathrm{i},\left(\begin{array}{rrr}-\mathrm{i} & 1 & 0 \\ -\mathrm{i} & 1 & -1 \\ 0 & 0 & -\mathrm{i}\end{array}\right) ;(\mathrm{g})-2,\left(\begin{array}{rrrr}1 & 0 & -1 & 1 \\ 0 & 1 & 0 & 1 \\ -1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1\end{array}\right) ;(h) 1,\left(\begin{array}{rrrrr}1 & -1 & -1 & -1 & -1 \\ 0 & 1 & -1 & -1 & -1 \\ 0 & 0 & 1 & -1 & -1 \\ 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 & 1\end{array}\right)$.

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Problem 1

Find all invariant subspaces $W \subset \mathbb{R}^2$ of the following linear transformations $L: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ :
(a) the scaling transformation $(2 x, 3 y, 4 z)^T$;
(b) the shear $(x+3 y, y, z)^T$;
(c) counterclockwise rotation by a $45^{\circ}$ angle around the $x$-axis.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 1

Find the eigenvalues and an orthonormal eigenvector basis for the following symmetric matrices:
(a) $\left(\begin{array}{rr}2 & 6 \\ 6 & -7\end{array}\right)$
(b) $\left(\begin{array}{rr}5 & -2 \\ -2 & 5\end{array}\right)$,
(c) $\left(\begin{array}{rr}2 & -1 \\ -1 & 5\end{array}\right)$,
(d) $\left(\begin{array}{lll}1 & 0 & 4 \\ 0 & 1 & 3 \\ 4 & 3 & 1\end{array}\right)$,
(e) $\left(\begin{array}{rrr}6 & -4 & 1 \\ -4 & 6 & -1 \\ 1 & -1 & 11\end{array}\right)$.

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Problem 1

Establish a Schur Decomposition for the matrices (a) $\left(\begin{array}{rr}1 & -1 \\ 1 & 3\end{array}\right)$,
(b) $\left(\begin{array}{rr}1 & -2 \\ -2 & 1\end{array}\right)$,
(c) $\left(\begin{array}{rr}8 & 9 \\ -6 & -7\end{array}\right)$,
(d) $\left(\begin{array}{rr}1 & 5 \\ -2 & -1\end{array}\right)$
(e) $\left(\begin{array}{rrr}2 & -1 & 2 \\ -2 & 3 & -1 \\ -6 & 6 & -5\end{array}\right)$,
(f) $\left(\begin{array}{rrr}0 & 2 & -1 \\ -1 & -1 & 1 \\ -1 & 0 & 0\end{array}\right)$.

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Problem 1

Find the singular values of the following matrices:
(a) $\left(\begin{array}{ll}1 & 1 \\ 0 & 2\end{array}\right)$,
(b) $\left(\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right)$,
(c) $\left(\begin{array}{rr}1 & -2 \\ -3 & 6\end{array}\right)$,
(d) $\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 3 & 0\end{array}\right)$,
(e) $\left(\begin{array}{rrrr}2 & 1 & 0 & -1 \\ 0 & -1 & 1 & 1\end{array}\right)$,
(f) $\left(\begin{array}{rrr}1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1\end{array}\right)$.

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03:04

Problem 1

Find the mean, the variance, and the standard deviation of the following data sets:
(a) $1.1,1.3,1.5,1.55,1.6,1.9,2,2.1$; (b) $2 ., .9, .7,1.5,2.6,3, .8,1.4$; (c) $-2.9,-.5, .1,-1.5$, $-3.6,1.3, .4,-.7$; (d) $1.1, .2, .1, .6,1.3,-4,-.1, .4 ;$ (e) $.9,-.4,-.8, . .1 .,-1.6,-1.2,-.7$.

Jerelyn Nevil
Jerelyn Nevil
Numerade Educator
02:30

Problem 2

Suppose a radioactive material has a half-life of 100 years. What is the decay rate $\gamma$ ? Starting with an initial sample of 100 grams, how much will be left after 10 years? after 100 years? after 1,000 years?

Taylor Shimono
Taylor Shimono
Numerade Educator
05:04

Problem 2

(a) Find the eigenvalues of the rotation matrix $R_\theta=\left(\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$. For what values of $\theta$ are the eigenvalues real?
(b) Explain why your answer gives an immediate solution to Exercise 1.5.7c.

Shelby Mohamed
Shelby Mohamed
Numerade Educator

Problem 2

Find the eigenvalues and a basis for the each of the eigenspaces of the following matrices. Which are complete?
(a) $\left(\begin{array}{rr}4 & -4 \\ 1 & 0\end{array}\right)$
(b) $\left(\begin{array}{ll}6 & -8 \\ 4 & -6\end{array}\right)$,
(c) $\left(\begin{array}{ll}3 & -2 \\ 4 & -1\end{array}\right)$,
(d) $\left(\begin{array}{rr}\mathrm{i} & -1 \\ 1 & \mathrm{i}\end{array}\right)$,
(e) $\left(\begin{array}{rrr}4 & -1 & -1 \\ 0 & 3 & 0 \\ 1 & -1 & 2\end{array}\right)$,
(f) $\left(\begin{array}{rrr}-6 & 0 & -8 \\ -4 & 2 & -4 \\ 4 & 0 & 6\end{array}\right)$,
(g) $\left(\begin{array}{rrr}-2 & 1 & -1 \\ 5 & -3 & 6 \\ 5 & -1 & 4\end{array}\right)$,
(h) $\left(\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ -1 & 1 & -1 & 0 \\ 1 & 0 & -1 & 0\end{array}\right)$,
(i) $\left(\begin{array}{rrrr}-1 & 0 & 1 & 2 \\ 0 & 1 & 0 & 1 \\ -1 & -4 & 1 & -2 \\ 0 & 1 & 0 & 1\end{array}\right)$.

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02:05

Problem 2

Find all invariant subspaces of the following matrices:
(a) $\left(\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right)$,
(b) $\left(\begin{array}{rr}3 & 0 \\ -2 & 3\end{array}\right)$,
(c) $\left(\begin{array}{lll}0 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 0 & 0\end{array}\right)$,
(d)
$\left(\begin{array}{rrr}-6 & 0 & -8 \\ -4 & 2 & -4 \\ 4 & 0 & 6\end{array}\right)$
(e) $\left(\begin{array}{llll}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right)$,
(f) $\left(\begin{array}{llll}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1\end{array}\right)$.

Victor Salazar
Victor Salazar
Numerade Educator
02:18

Problem 2

Determine whether the following symmetric matrices are positive definite by computing their eigenvalues. Validate your conclusions by using the methods from Chapter 5.
(a) $\left(\begin{array}{rr}2 & -2 \\ -2 & 3\end{array}\right)$
(b) $\left(\begin{array}{rr}-2 & 3 \\ 3 & 6\end{array}\right)$,
(c) $\left(\begin{array}{rrr}1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}4 & -1 & -2 \\ -1 & 4 & -1 \\ -2 & -1 & 4\end{array}\right)$.

Carole Wastog
Carole Wastog
Numerade Educator
00:59

Problem 2

Show that a real unitary matrix is an orthogonal matrix.

Raj Bala
Raj Bala
Numerade Educator
02:53

Problem 2

Write out the singular value decomposition (8.52) of the matrices in Exercise 8.7.1.

Charles Carter
Charles Carter
Numerade Educator
01:39

Problem 2

Find the mean, the variance, and the standard deviation of the data sets $\{f(x) \mid x=i / 10, i=-10, \ldots, 10\}$ associated with the following functions $f(x)$ :
(a) $3 x+1$,
(b) $x^2$,
(c) $x^3-2 x$,
(d) $e^{-x}$,
(e) $\tan ^{-1} x$.

Ahmad Reda
Ahmad Reda
Numerade Educator
01:04

Problem 3

Carbon-14 has a half-life of 5730 years. Human skeletal fragments discovered in a cave are analyzed and found to have only $6.24 \%$ of the carbon- 14 that living tissue would have. How old are the remains?

David Collins
David Collins
Numerade Educator
01:27

Problem 3

Answer Exercise 8.2.2a for the reflection matrix $F_\theta=\left(\begin{array}{rr}\cos \theta & \sin \theta \\ \sin \theta & -\cos \theta\end{array}\right)$.

James Kiss
James Kiss
Numerade Educator

Problem 3

Which of the following matrices admit eigenvector bases of $\mathbb{R}^n$ ? For those that do, exhibit such a basis. If not, what is the dimension of the subspace of $\mathbb{R}^n$ spanned by the eigenvectors?
(a) $\left(\begin{array}{ll}1 & 3 \\ 3 & 1\end{array}\right)$
(b) $\left(\begin{array}{rr}1 & 3 \\ -3 & 1\end{array}\right)$,
(c) $\left(\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}1 & -2 & 0 \\ 0 & -1 & 0 \\ 4 & -4 & -1\end{array}\right)$,
(e) $\left(\begin{array}{rrr}1 & -2 & 0 \\ 0 & -1 & 0 \\ 0 & -4 & -1\end{array}\right)$.
(f) $\left(\begin{array}{rrr}2 & 0 & 0 \\ 1 & -1 & 1 \\ 2 & 1 & -1\end{array}\right)$,
(g) $\left(\begin{array}{rrr}0 & 0 & -1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right)$.
(h) $\left(\begin{array}{rrrr}0 & 0 & -1 & 1 \\ 0 & -1 & 0 & 1 \\ 1 & 0 & -1 & 0 \\ 1 & 0 & 1 & 0\end{array}\right)$.

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Problem 3

Find all complex invariant subspaces and all real invariant subspaces of
(a) $\left(\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right)$,
(b) $\left(\begin{array}{rr}-1 & 2 \\ 1 & 1\end{array}\right)$,
(c) $\left(\begin{array}{rrr}3 & 3 & 5 \\ 5 & 6 & 5 \\ -5 & -8 & -7\end{array}\right)$,
(d) $\left(\begin{array}{rrr}2 & 5 & 5 \\ 0 & 2 & 0 \\ 0 & -5 & -3\end{array}\right)$.

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Problem 3

Prove that a symmetric matrix is negative definite if and only if all its eigenvalues are negative.

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Problem 3

Prove Proposition 8.44.

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Problem 3

(a) Construct the singular value decomposition of the shear matrix $A=\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right)$.
(b) Explain how a shear can be realized as a combination of a rotation, and a stretch, followed by a second rotation.

Victor Salazar
Victor Salazar
Numerade Educator
01:36

Problem 3

Determine the variance and standard deviation of the normally distributed data points
$$
\left\{e^{-x^2 / \sigma} \mid x=i / 10, i=-10, \ldots, 10\right\} \text { for } \sigma=1,2, \text { and } 10 .
$$

Bon Zapata
Bon Zapata
Numerade Educator
01:33

Problem 4

Prove that if $t_*$ is the half-life of a radioactive material, then $u\left(n t_*\right)=2^{-n} u(0)$. Explain the meaning of this equation in your own words.

Sid Wan
Sid Wan
University of Louisville
12:55

Problem 4

Write down (a) a $2 \times 2$ matrix that has 0 as one of its eigenvalues and $(1,2)^T$ as a corresponding eigenvector; (b) a $3 \times 3$ matrix that has $(1,2,3)^T$ as an eigenvector for the eigenvalue -1 .

Chris Trentman
Chris Trentman
Numerade Educator
00:34

Problem 4

Answer Exercise 8.3.3 with $\mathbb{R}^n$ replaced by $\mathbb{C}^n$.

Brandon Fox
Brandon Fox
Numerade Educator
02:37

Problem 4

Prove that if $W$ is an invariant subspace for $A$, then it is also invariant for $A^2$. Is the converse to this statement valid?

ET
Ed Tam
Numerade Educator
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Problem 4

How many orthonormal eigenvector bases does a symmetric $n \times n$ matrix have?

Michelle Z.
Michelle Z.
Numerade Educator

Problem 4

Write out a new proof of the Spectral Theorem 8.38 based on the Schur Decomposition.

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04:19

Problem 4

Find the condition number of the following matrices. Which would you characterize as ill-conditioned?
(a) $\left(\begin{array}{rr}2 & -1 \\ -3 & 1\end{array}\right)$,
(b) $\left(\begin{array}{rr}-.999 & .341 \\ -1.001 & .388\end{array}\right)$,
(c) $\left(\begin{array}{ll}1 & 2 \\ 1.001 & 1.9997\end{array}\right)$,
(d) $\left(\begin{array}{rrr}-1 & 3 & 4 \\ 2 & 10 & 6 \\ 1 & 2 & -3\end{array}\right)$,
(e) $\left(\begin{array}{rrr}72 & 96 & 103 \\ 42 & 55 & 59 \\ 67 & 95 & 102\end{array}\right)$,
(f) $\left(\begin{array}{rrrr}5 & 7 & 6 & 5 \\ 7 & 10 & 8 & 7 \\ 6 & 8 & 10 & 9 \\ 5 & 7 & 9 & 10\end{array}\right)$.

Supratim Pal
Supratim Pal
Numerade Educator
07:23

Problem 4

Prove that $\sigma_{x y}=\overline{I y}-\mathbf{T} \bar{y}$, where $\bar{x}$ and $\bar{y}$ are the means of $\left\{x_i\right\}$ and $\left\{y_i\right\}$, respectively, while $\overline{x y}$ denotes the mean of the product variable $\left\{x_i y_i\right\}$.

Abhirup Pal
Abhirup Pal
Numerade Educator
05:44

Problem 5

A bacteria colony grows according to the equation $d u / d t=1.3 u$. How long until the colony doubles? quadruples? If the initial population is 2 , how long until the population reaches 2 million?

Kader Bayar
Kader Bayar
Numerade Educator
05:17

Problem 5

(a) Write out the characteristic equation for the matrix $\left(\begin{array}{ccc}0 & 1 & 0 \\ 0 & 0 & 1 \\ \alpha & \beta & \gamma\end{array}\right)$.
(b) Show that, given any 3 numbers $a, b$, and $c$, there is a $3 \times 3$ matrix with characteristic equation $-\lambda^3+a \lambda^2+b \lambda+c=0$.

Anthony Ramos
Anthony Ramos
Numerade Educator
12:18

Problem 5

(a) Give an example of a $3 \times 3$ matrix with 1 as its only eigenvalue, and only one linearly independent eigenvector. (b) Find one that has two linearly independent eigenvectors.

Tim Strang
Tim Strang
Numerade Educator
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Problem 5

Let $V \subset \mathbb{R}^n$ be an invariant subepace for the $n \times n$ matrix $A$. Explain why every eigenvalue and eigenvector of the linear map obtained by restricting $A$ to $V$ are also eigenvalues and eigenvectors of $A$ itself.

Nick Johnson
Nick Johnson
Numerade Educator
03:06

Problem 5

Let $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$.
(a) Write down necessary and sufficient conditions on the entries $a, b, c, d$ that ensures that $A$ has only real eigenvalues.
(b) Verify that all symmetric $2 \times 2$ matrices satisfy your conditions.
(c) Write down a non-symmetric matrix that satisfies your conditions.

Chris Trentman
Chris Trentman
Numerade Educator
01:25

Problem 5

A complex matrix $A$ is called normal if it commutes with its Hermitian transpose: $A^{\dagger} A=A A^{\dagger}$. (a) Show that every real symmetric matrix is normal. (b) Show that every unitary matrix is normal. (c) Show that every real orthogonal matrix is normal. (d) Show that an upper triangular matrix is normal if and only if it is diagonal. (e) Show that the eigenvectors of a normal matrix form an orthogonal basis of $\mathrm{C}^n$ under the Hermitian dot product. $(f)$ Show that the converse is true: a matrix has an orthogonal eigenvector basis of $\mathbb{C}^n$ if and only if it is normal. Hint: Use the Schur Decomposition. $(g)$ How can you tell when a real matrix has a real orthonormal eigenvector basis?

Manik Pulyani
Manik Pulyani
Numerade Educator
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Problem 5

Find the closest rank 1 and rank 2 matrices to the matrices in Exercise 8,7.4.

Michelle Z.
Michelle Z.
Numerade Educator
03:08

Problem 5

Show that one can compute the variance of a set of measurements without reference to the mean by the following formula
$$
\sigma_x^2=\frac{\nu}{2 m} \sum_{i=1}^m \sum_{j=1}^m\left(x_i-x_j\right)^2=\frac{\nu}{m} \sum_{i<j}\left(x_i-x_j\right)^2 .
$$

AN
Anjali Nagulpally
Numerade Educator
04:23

Problem 6

Deer in northern Minnesota reproduce according to the linear differential equation $\frac{d u}{d t}=.27 u$ where $t$ is measured in years. If the initial population is $u(0)=5,000$ and the environment can sustain at most 1,000,000 deer, how long until the deer run out of resources?

Himanshu Kushwaha
Himanshu Kushwaha
Numerade Educator
03:35

Problem 6

Find the eigenvalues and eigenvectors of the cross product matrix $A=\left(\begin{array}{rrr}0 & c & -b \\ -c & 0 & a \\ b & -a & 0\end{array}\right)$.

Shelby Mohamed
Shelby Mohamed
Numerade Educator
03:33

Problem 6

True or false: (a) Every diagonal matrix is complete.
(b) Every upper triangular matrix is complete.

Patrick Burns
Patrick Burns
Numerade Educator
03:59

Problem 6

True or false: If $V$ and $W$ are invariant subspaces for the matrix $A$, then so is
(a) $V+W$
(b) $V \cap W$;
(c) $V \cup W$;
(d) $V \backslash W$.

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator
07:34

Problem 6

Let $A^T=-A$ be a real, skew-symmetric $n \times n$ matrix. (a) Prove that the only possible real eigenvalue of $A$ is $\lambda=0$. (b) More generally, prove that all eigenvalues $\lambda$ of $A$ are purely imaginary, i.e., $\operatorname{Re} \lambda=0$. (c) Explain why 0 is an eigenvalue of $A$ whenever $n$ is odd. (d) Explain why, if $n=3$, the eigenvalues of $A \neq 0$ are $0, i \omega,-i \omega$, for some real $\omega \neq 0$. (e) Verify these facts for the particular matrices
(i) $\left(\begin{array}{rr}0 & -2 \\ 2 & 0\end{array}\right)$,
(ii) $\left(\begin{array}{rrr}0 & 3 & 0 \\ -3 & 0 & -4 \\ 0 & 4 & 0\end{array}\right)$,
(iii) $\left(\begin{array}{rrr}0 & 1 & -1 \\ -1 & 0 & -1 \\ 1 & 1 & 0\end{array}\right)$,
(kv) $\left(\begin{array}{rrrr}0 & 0 & 2 & 0 \\ 0 & 0 & 0 & -3 \\ -2 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0\end{array}\right)$.

Chris Trentman
Chris Trentman
Numerade Educator

Problem 6

For each of the following Jordan matrices, identify the Jordan blocks. Write down the eigenvalues, the eigenvectors, and the Jordan basis. Clearly identify the Jordan chains.
(a) $\left(\begin{array}{ll}2 & 1 \\ 0 & 2\end{array}\right)$,
(b) $\left(\begin{array}{rr}-3 & 0 \\ 0 & 6\end{array}\right)$,
(c) $\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right)$,
(d) $\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right)$,
(e) $\left(\begin{array}{llll}4 & 0 & 0 & 0 \\ 0 & 3 & 1 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 2\end{array}\right)$

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11:13

Problem 6

Solve the following systems of equations using Gaussian Elimination with three-digit rounding arithmetic. Is your answer a reasonable approximation to the exact solution? Compare the accuracy of your answers with the condition number of the coefficient matrix, and discuss the implications of ill-conditioning.
(a)
$$
\begin{aligned}
1000 x+999 y & =1 \\
554 x+555 y & =-1
\end{aligned}
$$
$$
97 x+175 y+83 z=1, \quad 3.001 x+2.999 y+5 z=1,
$$
(b)
$$
\begin{aligned}
& 44 x+78 y+37 z=1, \\
& 52 x+97 y+46 z=1 .
\end{aligned}
$$
(c)
$$
\begin{aligned}
-x+1.002 y-2.999 z & =2 \\
2.002 x+4 y+2 z & =1.002 .
\end{aligned}
$$

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
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Problem 6

Let $A$ be an $m \times n$ matrix that is normalized, meaning that each of its column sums is zero. Show that $A B$, where $B$ is any $n \times k$ matrix, is also normalized.

Victor Salazar
Victor Salazar
Numerade Educator
01:34

Problem 7

Consider the inhomogeneous differential equation $\frac{d u}{d t}=a u+b$, where $a, b$ are constants.
(a) Show that $u_{\star}=-b / a$ is a constant equilibrium solution. (b) Solve the differential equation. Hint: Look at the differential equation satisfied by $v=u-u_*$.
(c) Discuss the stability of the equilibrium solution $u_{\star}$ *

Anand Jangid
Anand Jangid
Numerade Educator
01:00

Problem 7

Find all eigenvalues and eigenvectors of the following complex matrices:
(a) $\left(\begin{array}{cc}\mathrm{i} & 1 \\ 0 & -1+\mathrm{i}\end{array}\right)$,
(b) $\left(\begin{array}{rr}2 & i \\ -i & -2\end{array}\right)$,
(c) $\left(\begin{array}{rr}i-2 & i+1 \\ i+2 & i-1\end{array}\right)$
(d)
$$
\left(\begin{array}{crr}
1+i & -1-i & 1-i \\
2 & -2-i & 2-2 i \\
-1 & 1+i & -1+2 i
\end{array}\right)
$$

Raj Bala
Raj Bala
Numerade Educator
04:42

Problem 7

Prove that if $A$ is a complete matrix, then so is $c A+d \mathrm{I}$, where $c, d$ are any scalars.

Elham Kordzadeh
Elham Kordzadeh
Numerade Educator

Problem 7

True or false: If $V$ is an invariant subspace for the $n \times n$ matrix $A$ and $W$ is an invariant subspace for the $n \times n$ matrix $B$, then $V+W$ is an invariant subspace for the matrix $A+B$.

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04:40

Problem 7

(a) Prove that every eigenvalue of a Hermitian matrix $A$, satisfying $A^T=\bar{A}$ as in Exercise 3.6.45, is real. (b) Show that the eigenvectors corresponding to distinct eigenvalues are orthogonal under the Hermitian dot product on $\mathrm{C}^n$. (c) Find the eigenvalues and eigenvectors of the following Hermitian matrices, and verify orthogonality:
(i) $\left(\begin{array}{rr}2 & \mathrm{i} \\ -\mathrm{i} & -2\end{array}\right)$,
(ii) $\left(\begin{array}{cc}3 & 2-\mathrm{i} \\ 2+\mathrm{i} & -1\end{array}\right)$,
(iii) $\left(\begin{array}{rrr}0 & \mathrm{i} & 0 \\ -\mathrm{i} & 0 & \mathrm{i} \\ 0 & -\mathrm{i} & 0\end{array}\right)$.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 7

Find a Jordan basis and Jordan canonical form for each of the following matrices:
(a) $\left(\begin{array}{ll}2 & 3 \\ 0 & 2\end{array}\right)$,
(b) $\left(\begin{array}{rr}-1 & -1 \\ 4 & -5\end{array}\right)$,
(c) $\left(\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}-3 & 1 & 0 \\ 1 & -3 & -1 \\ 0 & 1 & -3\end{array}\right)$,
(e) $\left(\begin{array}{rrr}-1 & 1 & 1 \\ -2 & -2 & -2 \\ 1 & -1 & -1\end{array}\right)$,
(f) $\left(\begin{array}{rrrr}2 & -1 & 1 & 2 \\ 0 & 2 & 0 & 1 \\ 0 & 0 & 2 & -1 \\ 0 & 0 & 0 & 2\end{array}\right)$.

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00:51

Problem 7

(a) Compute the singular values and condition numbers of the $2 \times 2,3 \times 3$, and $4 \times 4$ Hilbert matrices. (b) What is the smallest Hilbert matrix with condition number larger than $10^6$ ?

AG
Ankit Gupta
Numerade Educator
13:03

Problem 7

Given a singular value decomposition $A=P \Sigma Q^T$, prove that if the columns of $A$ have zero mean, then so do the columns of $P \Sigma$.

Andrija Isakov
Andrija Isakov
Numerade Educator
02:34

Problem 8

Use the method of Exercise 8.1.7 to solve the following initial value problems:
(a) $\frac{d u}{d t}=2 u-1, u(0)=1$, (b) $\frac{d u}{d t}=5 u+15, u(1)=-3$, (c) $\frac{d u}{d t}=-3 u+6, u(2)=-1$.

Kyler Gray
Kyler Gray
Numerade Educator
02:23

Problem 8

Find all eigenvalues and eigenvectors of
(a) the $n \times n$ zero matrix $\mathrm{O}$;
(b) the $n \times n$ identity matrix $\mathrm{I}$.

Minh Le
Minh Le
Numerade Educator
03:02

Problem 8

(a) Prove that if $A$ is complete, then so is $A^2$.
(b) Give an example of an incomplete matrix $A$ such that $A^2$ is complete.

Andrija Isakov
Andrija Isakov
Numerade Educator
07:15

Problem 8

True or false: If $W$ is an invariant subspace of the matrix $A$, then it is also an invariant subspace of $A^T$.

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator
02:04

Problem 8

Let $K, M$ be $n \times n$ matrices, with $M>0$ positive definite. A nonzero vector $\mathbf{v} \neq \mathbf{0}$ is called a generalized eigenvector of matrix pair $K, M$ if it satisfies the generalized eigenvalue equation
$$
K \mathbf{v}=\lambda M \mathbf{v}, \quad \mathbf{v} \neq \mathbf{0},
$$
where the scalar $\lambda$ is the corresponding generalized eigenvalue. Note that ordinary eigenvalue/eigenvectors of $K$ are when $M=1$.(a) Prove that $\lambda$ is a generalized eigenvalue if and only if it satisfies the generalized characteristic equation $\operatorname{det}(K-\lambda M)=0$.
(b) Prove that $\lambda$ is a generalized eigenvalue of the matrix pair $K, M$ if and only if it is an ordinary eigenvalue of the matrix $M^{-1} K$. How are the eigenvectors related? (c) Now suppose $K$ is a symmetric matrix. Prove that its generalized eigenvalues are all real.
Hint: First explain why this does not follow from part (a). Instead mimic the proof of part (a) of Theorem 8.32 , using the weighted Hermitian inner product $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T M \mathbf{w}$ in place of the dot product. (d) Show that if $K>0$, then its generalized eigenvalues are all positive: $\lambda>0$. (e) Prove that the eigenvectors corresponding to different generalized eigenvalues are orthogonal under the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T M \mathbf{w}$. (f) Show
Cuibuag Thascoag com
that, if the matrix pair $K, M$ has $n$ distinct generalized eigenvalues, then the eigenvectors form an orthogonal basis for $\mathbb{R}^n$. Remark. One can, by mimicking the proof of part $(c)$ of Theorem 8.32 , show that this holds even when there are repeated generalized eigenvalues.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:25

Problem 8

Write down all possible $4 \times 4$ Jordan matrioes that have only $\lambda=2$ as an eigenvalue.

Manik Pulyani
Manik Pulyani
Numerade Educator
02:09

Problem 8

(a) What are the singular values of a $1 \times n$ matrix? (b) Write down its singular value decomposition. (c) Write down its pseudoinverse.

Runpeng Li
Runpeng Li
Numerade Educator
08:31

Problem 8

Construct the $5 \times 5$ covariance matrix for the 5 data sets in Exercise 8.8 .1 and find its principal variances and principal directions. What do you think is the dimension of the subspace the data lies in?

Abhirup Pal
Abhirup Pal
Numerade Educator
01:53

Problem 9

The radioactive waste from a nuclear reactor has a half-life of 1000 years. Waste is continually produced at the rate of 5 tons per year and stored in a dump site.
(a) Set up an inhomogeneous differential equation, of the form in Exercise 8.1.7, to model the amount of radioactive waste. (b) Determine whether the amount of radioactive material at the dump increases indefinitely, decreases to zero, or eventually stabilizes at some fixed amount. (c) Starting with a brand new site, how long will it be until the dump contains 100 tons of radioactive material?

Joseph Liao
Joseph Liao
Numerade Educator
02:30

Problem 9

Find the eigenvalues and eigenvectors of an $n \times n$ matrix with every entry equal to 1 .
Hint: Try with $n=2,3$, and then generalize.

Minh Le
Minh Le
Numerade Educator
06:07

Problem 9

Let $U$ be an upper triangular matrix with all its diagonal entries equal. Prove that $U$ is complete if and only if $U$ is a diagonal matrix.

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
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Problem 9

True or false: If $W$ is an invariant subspace of the nonsingular matrix $A$, then it is also an invariant subspace of $A^{-1}$.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 9

Compute the generalized eigenvalues and eigenvectors, as in (8.34), for the following matrix pairs. Verify orthogonality of the eigenvectors under the appropriate inner product.
(a) $K=\left(\begin{array}{rr}3 & -1 \\ -1 & 2\end{array}\right), M=\left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right)$.
(b) $K=\left(\begin{array}{ll}3 & 1 \\ 1 & 1\end{array}\right), M=\left(\begin{array}{ll}2 & 0 \\ 0 & 1\end{array}\right)$,
(c) $K=\left(\begin{array}{rr}2 & -1 \\ -1 & 4\end{array}\right), M=\left(\begin{array}{rr}2 & -1 \\ -1 & 1\end{array}\right)$,
(d) $K=\left(\begin{array}{rrr}6 & -8 & 3 \\ -8 & 24 & -6 \\ 3 & -6 & 99\end{array}\right), M=\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 9\end{array}\right)$,
(e) $K=\left(\begin{array}{lll}1 & 2 & 0 \\ 2 & 8 & 2 \\ 0 & 2 & 1\end{array}\right), M=\left(\begin{array}{lll}1 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 1 & 1\end{array}\right),(f) K=\left(\begin{array}{rrr}5 & 3 & -5 \\ 3 & 3 & -1 \\ -5 & -1 & 9\end{array}\right), M=\left(\begin{array}{rrr}3 & 2 & -3 \\ 2 & 2 & -1 \\ -3 & -1 & 5\end{array}\right)$.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 9

Write down all $3 \times 3$ Jordan matrices that have eigenvalues 2 and 5 (and no others).

Victor Salazar
Victor Salazar
Numerade Educator
00:52

Problem 9

Answer Exercise 8.7 .8 for an $m \times 1$ matrix.

James Kiss
James Kiss
Numerade Educator

Problem 9

For each of the following subsets $S \subset \mathbb{R}^3$. (i) Compute a fairly dense sample of data points $z_i \in S$; (ii) find the principal components of your data set, using $\mu=.95$ in the criterion in (8.78); (iii) using your principal components, estimate the dimension of the set $S$. Does your estimate coincide with the actual dimension? If not, explain any discrepancies.
(a) The line segment $S=\left\{(t+1,3 t-1,-2 t)^T \mid-1 \leq t \leq 1\right\}$;
(b) the set of points $z$ on the three coordinate axes with Euclidean norm $\| z \mid \leq 1$;
(c) the set of "probability vectors" $S=\left\{(x, y, z)^T \mid 0 \leq x, y, z \leq 1, x+y+z=1\right\}$;
(d) the unit ball $S=\{\|z\| \leq 1\}$ for the Euclidean norm;
(e) the unit sphere $S=\{\|z\|=1\}$ for the Euclidean norm;
(f) the unit ball $S=\left\{\mid z \|_{\infty} \leq 1\right\}$ for the $\infty$ norm;
(g) the unit sphere $S=\left\{\|z\|_{\infty}=1\right\}$ for the $\infty$ norm.

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01:46

Problem 10

Suppose that hunters are allowed to shoot a fixed number of the northern Minnesota deer in Exercise 8.1.6 each year. (a) Explain why the population model takes the form $\frac{d u}{d t}=.27 u-b$, where $b$ is the number killed yearly. (Ignore the seasonal aspects of hunting.)
(b) If $b=1,000$, how long until the deer run out of resources? Hint: See Exercise 8.1.7.
(c) What is the maximal rate at which deer can be hunted without causing their extinction?

James Kiss
James Kiss
Numerade Educator
05:35

Problem 10

Let $A$ be a given square matrix. (a) Explain in detail why every nonzero scalar multiple of an eigenvector of $A$ is also an eigenvector. (b) Show that every nonzero linear combination of two eigenvectors $\mathbf{v}, \mathbf{w}$ corresponding to the same eigenvalue is also an eigenvector. (c) Prove that a linear combination $c \mathbf{v}+d \mathbf{w}$, with $c, d \neq 0$, of two eigenvectors corresponding to different eigenvalues is never an eigenvector.

Jacob Fry
Jacob Fry
Numerade Educator

Problem 10

Suppose $\mathbf{v}_1, \ldots, \mathbf{v}_n$ is an eigenvector basis for the complete matrix $A$, with $\lambda_1, \ldots, \lambda_n$ the corresponding eigenvalues. Prove that every eigenvalue of $A$ is one of the $\lambda_1, \ldots, \lambda_n$.

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Problem 10

Which $2 \times 2$ orthogonal matrices have a nontrivial real invariant subspace?

Nick Johnson
Nick Johnson
Numerade Educator
02:11

Problem 10

Let $L=L^*: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a self-adjoint linear transformation with respect to the inner product $\langle\cdot,-\rangle$. Prove that all its eigenvalues are real and the eigenvectors are orthogonal. Hint: Mimic the proof of Theorem 8.32, replacing the dot product by the given inner product.

Nick Johnson
Nick Johnson
Numerade Educator
01:13

Problem 10

Write down a formula for the inverse of a Jordan block matrix.
Hint: Try some small examples first to help in figuring out the pattern.

Chandra Jain
Chandra Jain
Numerade Educator
01:07

Problem 10

True or false: Every matrix has at least one singular value.

Kaitlyn Wu
Kaitlyn Wu
Numerade Educator
06:47

Problem 10

Using the Euclidean norm, compute a fairly dense sample of points on the unit sphere $S=\left\{\mathbf{x} \in \mathbb{R}^3 \mid\|\mathbf{x}\|=1\right\}$. (a) Set $\mu=.95$ in (8.78), and then find the principal components of your data set. Do they indicate the two-dimensional nature of the sphere? If not, why not? (b) Now look at the subset of your data that is within a distance $r>0$ of the north pole, i.e., $\left\|\mathbf{x}-(0,0,1)^T\right\| \leq r$, and compute its principal components. How small does $r$ need to be to reveal the actual dimension of $S$ ? Interpret your calculations.

Pawan Yadav
Pawan Yadav
Numerade Educator
01:17

Problem 11

(a) Write down the exact solution to the initial value problem $\frac{d u}{d t}=\frac{2}{7} u, u(0)=\frac{1}{3}$.
(b) Suppose you make the approximation $u(0)=.3333$. At what point does your solution differ from the true solution by 1 unit? by 1000 units? (c) Answer the same question if you also approximate the coefficient in the differential equation by $\frac{d u}{d t}=.2857 u$.

Mike Gaerlan
Mike Gaerlan
Numerade Educator
02:09

Problem 11

Let $\lambda$ be a real eigenvalue of the real $n \times n$ matrix $A$, and $\mathbf{v}_1, \ldots, \mathbf{v}_k$ a basis for the associated eigenspace $V_\lambda$. Suppose $\mathbf{w} \in \mathbb{C}^n$ is a complex eigenvector, so $A \mathbf{w}=\lambda \mathbf{w}$. Prove that $\mathbf{w}=c_1 \mathbf{v}_1+\cdots+c_k \mathbf{v}_k$ is a complex linear combination of the real eigenspace basis. Hint: Look at the real and imaginary parts of the eigenvector equation.

Runpeng Li
Runpeng Li
Numerade Educator
03:02

Problem 11

Show that if $A$ is complete, then every similar matrix $B=S^{-1} A S$ is also complete.

Andrija Isakov
Andrija Isakov
Numerade Educator
02:32

Problem 11

True or false: If $Q \neq \pm \mathrm{I}$ is a $4 \times 4$ orthogonal matrix, then $Q$ has no real invariant subspaces.

Victor Salazar
Victor Salazar
Numerade Educator
05:05

Problem 11

The difference map $\Delta: \mathbb{C}^n \rightarrow \mathbb{C}^n$ is defined as $\Delta=S-\mathrm{I}$, where $S$ is the shift map of Exercise 8.2.13. (a) Write down the matrix $D$ corresponding to $\Delta$. (b) Prove that the sampled exponential vectors $\boldsymbol{\omega}_0, \ldots, \boldsymbol{\omega}_{n-1}$ from (5.102) form an eigenvector basis of $D$. What are the eigenvalues? (c) Prove that $K=D^T D$ has the same eigenvectors as $D$. What are its eigenvalues? (d) Is $K$ positive definite? (e) According to Theorem 8.32 the eigenvectors of a symmetric matrix are real and orthogonal. Use this to explain the orthogonality of the sampled exponential vectors. But, why aren't they real?

Jack Chen
Jack Chen
Numerade Educator
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Problem 11

True or false: If $A$ is complete, every generalized eigenvector is an ordinary eigenvector.

Victor Salazar
Victor Salazar
Numerade Educator
01:55

Problem 11

Explain why the singular values of $A$ are the same as the nonzero eigenvalues of the positive definite square root matrix $S=\sqrt{A^T A}$, defined in Exercise 8.5.27.

Victor Salazar
Victor Salazar
Numerade Educator
03:37

Problem 11

Show that the first principal direction $\mathrm{q}_1$ can be characterized as the direction of the line that minimizes the sums of the squares of its distances to the data points.

Alayna Abraham
Alayna Abraham
Numerade Educator
01:24

Problem 12

Let $a$ be complex. Prove that $u(t)=c e^{a t}$ is the (complex) solution to our scalar ordinary differential equation (8.1). Describe the asymptotic behavior of the solution as $t \rightarrow \infty$, and the stability properties of the zero equilibrium solution.

Adam Deaton
Adam Deaton
Numerade Educator
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Problem 12

True or false: If $\mathbf{v}$ is a real eigenvector of a real matrix $A$, then a nonzero complex multiple $\mathbf{w}=c \mathbf{v}$ for $c \in \mathbb{C}$ is a complex eigenvector of $A$.

Victor Salazar
Victor Salazar
Numerade Educator
02:16

Problem 12

8.3.12. (a) Prove that if $\mathbf{x} \pm \mathrm{i} \mathbf{y}$ is a complex conjugate pair of eigenvectors of a real matrix $A$ corresponding to complex conjugate eigenvalues $\mu \pm \mathrm{i} \nu$ with $\nu \neq 0$, then $\mathbf{x}$ and $\mathbf{y}$ are linearly independent real vectors. (b) More generally, if $\mathbf{v}_j=\mathbf{x}_j \pm \mathrm{i} \mathbf{y}_j, j=1, \ldots, k$, are complex conjugate pairs of eigenvectors corresponding to distinct pairs of complex conjugate eigenvalues $\mu_j \pm \mathrm{i} \nu_j, \nu_j \neq 0$, then the real vectors $\mathbf{x}_1, \ldots, \mathbf{x}_k, \mathbf{y}_1, \ldots, \mathbf{y}_k$ are linearly independent. (c) Prove that if $A$ is complete, then there exists a basis of $\mathbb{R}^n$ consisting of its real eigenvectors and real and imaginary parts of its complex eigenvectors.

Nick Johnson
Nick Johnson
Numerade Educator
02:11

Problem 12

(a) Let $A$ be an $n \times n$ symmetric matrix, and let $\mathrm{v}$ be an eigenvector. Prove that its orthogonal complement under the dot product, namely, $V^{\perp}=\left\{\mathbf{w} \in \mathbb{R}^n \mid \mathbf{v}_1 \cdot \mathbf{w}=0\right\}$, is an invariant subspace. (b) More generally, prove that if $W \subset \mathbb{R}^n$ is an invariant subspace, then its orthogonal complement $W^{\perp}$, is also invariant.

Nick Johnson
Nick Johnson
Numerade Educator
01:58

Problem 12

An $n \times n$ circulant matrix has the form $C=\left(\begin{array}{cccccc}c_0 & c_1 & c_2 & c_3 & \cdots & c_{n-1} \\ c_{n-1} & c_0 & c_1 & c_2 & \cdots & c_{n-2} \\ c_{n-2} & c_{n-1} & c_0 & c_1 & \cdots & c_{n-3} \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ c_1 & c_2 & c_3 & c_4 & \cdots & c_0\end{array}\right)$, in which the entries of each succeeding row are obtained by moving all the previous row's entries one slot to the right, the last entry moving to the front. (a) Check that the shift matrix $A$ of Exercise 8.2.13, the difference matrix $D$, and its symmetric product $K=D^T D$ of Exercise 8.5.11 are all circulant matrices. (b) Prove that the sampled exponential vectors $\omega_0, \ldots, \omega_{n-1}, c f .(5.102)$, are eigenvectors of $C$. Thus, all circulant matrices have the same eigenvectors! What are the eigenvalues? (c) Prove that $F_n^{-1} C F_n=\Lambda$, where $F_n$ is the Fourier matrix in Exercise 5.6.9 and $\Lambda$ is the diagonal matrix with the eigenvalues of $C$ along the diagonal. (d) Find the eigenvalues and eigenvectors of the following circulant matrices:
(i) $\left(\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right)$,
(ii) $\left(\begin{array}{lll}1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1\end{array}\right)$,
(iii) $\left(\begin{array}{rrrr}1 & -1 & -1 & 1 \\ 1 & 1 & -1 & -1 \\ -1 & 1 & 1 & -1 \\ -1 & -1 & 1 & 1\end{array}\right)$,
(iv)
$\left(\begin{array}{rrrr}2 & -1 & 0 & -1 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ -1 & 0 & -1 & 2\end{array}\right)$
(e) Find the eigenvalues of the tricirculant matrices in Exercise 1.7.13. Can you find a general formula for the $n \times n$ version? Explain why the eigenvalues must be real and positive. Does your formula reflect this fact? (f) Which of the preceding matrices are invertible? Write down a general criterion for checking the invertibility of circulant matrices.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 12

True or false: Every generalized eigenvector belongs to a Jordan chain.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 12

Prove that if the square matrix $A$ is nonsingular, then the singular values of $A^{-1}$ are the reciprocals of the singular values of $A$. How are their condition numbers related?

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01:09

Problem 12

Let $\mathbf{x}_i=\left(x_i, y_i\right), i=1, \ldots, m$, be a set of data points in the plane. Suppose $L^* \subset \mathbb{R}^2$ is the line that minimizes the sums of the squares of the distances from the data points to it, i.e., $\operatorname{dist}(\mathbf{x}, L)=\sum_{i=1}^m \operatorname{dist}\left(\mathbf{x}_i, L\right)$, among all lines $L \subset \mathbb{R}^2$. (a) Prove that $\overline{\mathbf{x}}=(\bar{x}, \bar{y}) \in L^*$. (b) Use Exercise 8.8.11 to find $L^*$. (c) Apply your result to the data points in Example 5.14 and compare the resulting line $L^*$ with the least squares line that was found there.

Fuzail Shakir
Fuzail Shakir
Numerade Educator
03:34

Problem 13

(a) Prove that if $u_1(t)$ and $u_2(t)$ are any two distinct solutions to $\frac{d u}{d t}=a u$ with $a>0$. then $\left|u_1(t)-u_2(t)\right| \rightarrow \infty$ as $t \rightarrow \infty$. (b) If $a=.02$ and $u_1(0)=.1, u_2(0)=.05$, how long do you have to wait until $\left|u_1(t)-u_2(t)\right|>1,000$ ?

Christian Otero
Christian Otero
Numerade Educator
07:27

Problem 13

Define the shift map $S: \mathbb{C}^n \rightarrow \mathbb{C}^n$ by $S\left(v_1, v_2, \ldots, v_{n-1}, v_n\right)^T=\left(v_2, v_3, \ldots, v_n, v_1\right)^T$.
(a) Prove that $S$ is a linear map, and write down its matrix representation $A$.
(b) Prove that $A$ is an orthogonal matrix. (c) Prove that the sampled exponential vectors $\omega_0, \ldots, \omega_{n-1}$ defined in (5.102) form an eigenvector basis of $A$. What are the eigenvalues?

Anthony Ramos
Anthony Ramos
Numerade Educator
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Problem 13

Diagonalize the following matrices:
(a) $\left(\begin{array}{ll}3 & -9 \\ 2 & -6\end{array}\right)$,
(b) $\left(\begin{array}{ll}5 & -4 \\ 2 & -1\end{array}\right)$,
(c) $\left(\begin{array}{rr}-4 & -2 \\ 5 & 2\end{array}\right)$,
(d) $\left(\begin{array}{rrr}-2 & 3 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 3\end{array}\right)$.
(e) $\left(\begin{array}{rrr}8 & 0 & -3 \\ -3 & 0 & -1 \\ 3 & 0 & -2\end{array}\right)$,
(f) $\left(\begin{array}{rrr}3 & 3 & 5 \\ 5 & 6 & 5 \\ -5 & -8 & -7\end{array}\right)$,
(g) $\left(\begin{array}{rrr}2 & 5 & 5 \\ 0 & 2 & 0 \\ 0 & -5 & -3\end{array}\right)$,
(h) $\left(\begin{array}{rrrr}1 & 0 & -1 & 1 \\ 0 & 2 & -1 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -2\end{array}\right)$,
(i) $\left(\begin{array}{llll}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right)$,
(j) $\left(\begin{array}{rrrr}2 & 1 & -1 & 0 \\ -3 & -2 & 0 & 1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 1 & -1\end{array}\right)$.

Victor Salazar
Victor Salazar
Numerade Educator
06:03

Problem 13

Write out the spectral factorization of the following matrices:
(a) $\left(\begin{array}{rr}-3 & 4 \\ 4 & 3\end{array}\right)$,
(b) $\left(\begin{array}{rr}2 & -1 \\ -1 & 4\end{array}\right)$,
(c) $\left(\begin{array}{lll}1 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 1\end{array}\right)$
(d) $\left(\begin{array}{rrr}3 & -1 & -1 \\ -1 & 2 & 0 \\ -1 & 0 & 2\end{array}\right)$.

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 13

True or false: If $\mathbf{w}$ is a generalized eigenvector of $A$, then $\mathbf{w}$ is a generalized eigenvector of every power $A^j$, for $j \in \mathbb{N}$, thereof.

Victor Salazar
Victor Salazar
Numerade Educator
03:13

Problem 13

True or false: The singular values of $A^T$ are the same as the singular values of $A$.

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator
07:29

Problem 14

(a) Compute the eigenvalues and corresponding eigenvectors of $A=\left(\begin{array}{rrr}1 & 4 & 4 \\ 3 & -1 & 0 \\ 0 & 2 & 3\end{array}\right)$.
(b) Compute the trace of $A$ and check that it equals the sum of the eigenvalues. (c) Find the determinant of $A$ and check that it is equal to to the product of the eigenvalues.

Evelyn Huszar
Evelyn Huszar
Numerade Educator
00:47

Problem 14

Diagonalize the Fibonacci matrix $F=\left(\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right)$.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 14

Write out the spectral factorization of the matrices listed in Exercise 8.5.1.

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Problem 14

True or false: If $\mathbf{w}$ is a generalized eigenvector of index $k$ of $A$, then $\mathbf{w}$ is an ordinary eigenvector of $A^k$.

Victor Salazar
Victor Salazar
Numerade Educator
02:09

Problem 14

(a) Let $A$ be a nonsingular matrix. Prove that the product of the singular values of $A$ equals the absolute value of its determinant: $\sigma_1 \sigma_2 \cdots \sigma_n=|\operatorname{det} A|$.
(b) Does their sum equal the absolute value of the trace: $\sigma_1+\cdots+\sigma_n=|\operatorname{tr} A|$ ?
(c) Show that if $|\operatorname{det} A|<10^{-k}$, then its minimal singular value satisfies $\sigma_n<10^{-k / n}$.
(d) True or false: A matrix whose determinant is very small is ill-conditioned.
(e) Construct an ill-conditioned matrix with det $A=1$.

Jack Chen
Jack Chen
Numerade Educator
00:51

Problem 15

Verify the trace and determinant formulas $(8.25,26)$ for the matrices in Exercise 8.2.1.

Scott Stetson
Scott Stetson
Numerade Educator
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Problem 15

Diagonalize the matrix $\left(\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right)$ of rotation through $90^{\circ}$. How would you interpret the result? the result?

Victor Salazar
Victor Salazar
Numerade Educator
08:24

Problem 15

Construct a symmetric matrix with the following eigenvectors and eigenvalues, or explain why none exists:
(a) $\lambda_1=1$,
(b) $\lambda_1=-2, \mathbf{v}_1=(1,-1)^T$
$\lambda_2=-1, \quad \mathbf{v}_2=(-1,2)^T$,

Shelby Mohamed
Shelby Mohamed
Numerade Educator

Problem 15

Suppose you know all eigenvalues of a matrix as well as their algebraic and geometric multiplicities. Can you determine the matrix's Jordan canonical form?

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Problem 15

True or false: If $A$ is a symmetric matrix, then its singular values are the same as its eigenvalues.

Victor Salazar
Victor Salazar
Numerade Educator
02:49

Problem 16

(a) Find the explicit formula for the characteristic polynomial $\operatorname{det}(A-\lambda \mathrm{I})=-\lambda^3+a \lambda^2-b \lambda+c$ of a general $3 \times 3$ matrix. Verify that $a=\operatorname{tr} A$, $c=\operatorname{det} A$. What is the formula for $b$ ? (b) Prove that if $A$ has eigenvalues $\lambda_1, \lambda_2, \lambda_3$, then $a=\operatorname{tr} A=\lambda_1+\lambda_2+\lambda_3, b=\lambda_1 \lambda_2+\lambda_1 \lambda_3+\lambda_2 \lambda_3, c=\operatorname{det} A=\lambda_1 \lambda_2 \lambda_3$.

Tamara Worner
Tamara Worner
Numerade Educator
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Problem 16

Diagonalize the rotation matrices
(a) $\left(\begin{array}{rrr}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right)$
(b)
$$
\left(\begin{array}{ccc}
\frac{5}{13} & 0 & \frac{12}{13} \\
0 & 1 & 0 \\
-\frac{12}{13} & 0 & \frac{5}{13}
\end{array}\right)
$$

Victor Salazar
Victor Salazar
Numerade Educator
15:48

Problem 16

(a) Find the eigenvalues and eigenvectors of the matrix $A=\left(\begin{array}{rrr}2 & 1 & -1 \\ 1 & 2 & 1 \\ -1 & 1 & 2\end{array}\right)$.
(b) Use the eigenvalues to compute the determinant of $A$. (c) Is $A$ positive definite? Why or why not? (d) Find an orthonormal eigenvector basis of $\mathbb{R}^3$ determined by $A$ or explain why none exists. (e) Write out the spectral factorization of $A$ if possible. ( $f$ ) Use orthogonality to write the vector $(1,0,0)^T$ as a linear combination of eigenvectors of $A$.

Chris Trentman
Chris Trentman
Numerade Educator
00:54

Problem 16

True or false: If $\mathbf{w}_1, \ldots, \mathbf{w}_j$ is a Jordan chain for a matrix $A$, so are the scalar multiples $c \mathbf{w}_1, \ldots, c \mathbf{w}_j$ for all $c \neq 0$.

Victor Salazar
Victor Salazar
Numerade Educator
03:33

Problem 16

True or false: If $U$ is an upper triangular matrix whose diagonal entries are all positive, then its singular values are the same as its diagonal entries.

Patrick Burns
Patrick Burns
Numerade Educator

Problem 17

Prove that the eigenvalues of an upper triangular (or lower triangular) matrix are its diagonal entries.

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00:49

Problem 17

Which of these matrices have real diagonal forms?
(a) $\left(\begin{array}{rr}-2 & 1 \\ 4 & 1\end{array}\right)$,
(b) $\left(\begin{array}{rr}1 & 2 \\ -3 & 1\end{array}\right)$
(c) $\left(\begin{array}{rrr}0 & 1 & 0 \\ -1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right)$,
(d) $\left(\begin{array}{rrr}0 & 3 & 2 \\ -1 & 1 & -1 \\ 1 & -3 & -1\end{array}\right)$,
(e) $\left(\begin{array}{rrr}3 & -8 & 2 \\ -1 & 2 & 2 \\ 1 & -4 & 2\end{array}\right)$,
(f) $\left(\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -1 & -1 & -1 & -1\end{array}\right)$.

Siena Cizdziel
Siena Cizdziel
Numerade Educator
06:00

Problem 17

Use the spectral factorization to diagonalize the following quadratic forms:
(a) $x^2-3 x y+5 y^2$,
(b) $3 x^2+4 x y+6 y^2$,
(c) $x^2+8 x z+y^2+6 y z+z^2$,
(d) $\frac{3}{2} x^2-x y-x z+y^2+z^2$,
(e) $6 x^2-8 x y+2 x z+6 y^2-2 y z+11 z^2$.

AG
Ankit Gupta
Numerade Educator
02:23

Problem 17

Let $A$ and $B$ be $n \times n$ matrices. According to Exercise 8.2.23, the matrix products $A B$ and $B A$ have the same eigenvalues. Do they have the same Jordan form?

Tim Strang
Tim Strang
Numerade Educator
01:04

Problem 17

True or false: The singular values of $A^2$ are the squares $\sigma_i^2$ of the singular values of $A$.

Jameson Kuper
Jameson Kuper
Numerade Educator

Problem 18

Let $J_{a, n}$ be the $n \times n$ Jordan block matrix (8.23). Prove that its only eigenvalue is $\lambda=a$ and the only eigenvectors are the nonzero scalar multiples of the standard basis vector $\mathbf{e}_1$

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04:59

Problem 18

Diagonalize the following complex matrices:
(a) $\left(\begin{array}{ll}\mathrm{i} & 1 \\ 1 & \mathrm{i}\end{array}\right)$,
(b) $\left(\begin{array}{cc}1-i & 0 \\ \mathrm{i} & 2+\mathrm{i}\end{array}\right)$,
(c) $\left(\begin{array}{ll}2-\mathrm{i} & 2+\mathrm{i} \\ 3-\mathrm{i} & 1+\mathrm{i}\end{array}\right)$
(d) $\left(\begin{array}{rrr}-i & 0 & 1 \\ -i & 1 & -1 \\ 1 & 0 & -i\end{array}\right)$

Antonio Fonte
Antonio Fonte
Numerade Educator
02:41

Problem 18

Let $\mathbf{u}_1, \ldots, \mathbf{u}_n$ be an orthonormal basis of $\mathrm{R}^n$. Prove that it forms an eigenvector basis for some symmetric $n \times n$ matrix $A$. Can you characterize all such matrices?

Foster Wisusik
Foster Wisusik
Numerade Educator
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Problem 18

True or false: If the Jordan canonical form of $A$ is $J$, then that of $A^2$ is $J^2$.

Nick Johnson
Nick Johnson
Numerade Educator
03:33

Problem 18

True or false: If $B=S^{-1} A S$ are similar matrices, then $A$ and $B$ have the same singular values.

Patrick Burns
Patrick Burns
Numerade Educator
07:00

Problem 19

Suppose that $\lambda$ is an eigenvalue of $A$. (a) Prove that $c \lambda$ is an eigenvalue of the scalar multiple $c A$. (b) Prove that $\lambda+d$ is an eigenvalue of $A+d \mathbb{I}$. (c) More generally, $c \lambda+d$ is an eigenvalue of $B=c A+d \mathrm{I}$ for scalars $c, d$.

Chris Trentman
Chris Trentman
Numerade Educator

Problem 19

Write down a real matrix that has
(a) eigenvalues $-1,3$ and corresponding eigenvectors $\left(\begin{array}{r}-1 \\ 2\end{array}\right),\left(\begin{array}{l}1 \\ 1\end{array}\right)$,
(b) eigenvalues $0,2,-2$ and associated eigenvectors $\left(\begin{array}{r}-1 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{l}0 \\ 1 \\ 3\end{array}\right)$;
(c) an eigenvalue of 3 and corresponding eigenvectors $\left(\begin{array}{r}2 \\ -3\end{array}\right),\left(\begin{array}{l}1 \\ 2\end{array}\right)$;
(d) an eigenvalue $-1+2 \mathrm{i}$ and corresponding eigenvector $\left(\begin{array}{c}1+\mathrm{i} \\ 3 \mathrm{i}\end{array}\right)$;
(e) an eigenvalue -2 and corresponding eigenvector $\left(\begin{array}{r}2 \\ 0 \\ -1\end{array}\right)$;
(f) an eigenvalue $3+\mathrm{i}$ and corresponding eigenvector $\left(\begin{array}{c}1 \\ 2 \mathrm{i} \\ -1-\mathrm{i}\end{array}\right)$.

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Problem 19

True or false: A matrix with a real orthonormal eigenvector basis is symmetric.

Donna Densmore
Donna Densmore
Numerade Educator
11:41

Problem 19

(a) Give an example of a matrix $A$ such that $A^2$ has an eigenvector that is not an eigenvector of $A$. (b) Show that, in general, every eigenvalue of $A^2$ is the square of an eigenvalue of $A$.

Tim Strang
Tim Strang
Numerade Educator
03:00

Problem 19

Under the assumptions of Theorem 8.74, show that $\|A-B\|_2$ is also minimized when $B=A_k$ among all matrices with $\operatorname{rank} B \leq k$.

Chris Trentman
Chris Trentman
Numerade Educator
01:06

Problem 20

Show that if $\lambda$ is an eigenvalue of $A$, then $\lambda^2$ is an eigenvalue of $A^2$.

Nicholas Barvinok
Nicholas Barvinok
Numerade Educator
05:10

Problem 20

A matrix $A$ has eigenvalues -1 and 2 and associated eigenvectors $\left(\begin{array}{l}1 \\ 2\end{array}\right)$ and $\left(\begin{array}{l}2 \\ 3\end{array}\right)$.
Write down the matrix form of the linear transformation $L[\mathbf{u}]=A \mathbf{u}$ in terms of (a) the standard basis $\mathrm{e}_1, \mathbf{e}_2 ;(\mathrm{b})$ the basis consisting of its eigenvectors; (c) the basis $\left(\begin{array}{l}1 \\ 1\end{array}\right),\left(\begin{array}{l}3 \\ 4\end{array}\right)$.

R M
R M
Numerade Educator
03:58

Problem 20

Prove that every quadratic form can be written as $\mathbf{x}^T A \mathbf{x}=\|\mathbf{x}\|^2\left(\sum_{i=1}^n \lambda_i \cos ^2 \theta_i\right)$, where $\lambda_i$ are the eigenvalues of $A$ and $\theta_i=\Varangle\left(\mathbf{x}, \mathbf{v}_i\right)$ denotes the angle between $\mathbf{x}$ and the $i^{\text {th }}$ eigenvector.

Nicholas Barvinok
Nicholas Barvinok
Numerade Educator
01:28

Problem 20

(a) Prove that a Jordan block matrix $J_{0, n}$ with zero diagonal entries is nilpotent, as in Exercise 1.3.12. (b) Prove that a Jordan matrix is nilpotent if and only if all its diagonal entries are zero. (c) Prove that a matrix is nilpotent if and only if its Jordan canonical form is nilpotent. its only eigenvalue is 0 .
(d) Explain why a matrix is nilpotent if and only if

Nick Johnson
Nick Johnson
Numerade Educator

Problem 20

Suppose $A$ is an $m \times n$ matrix of rank $r<n$. Prove that there exist arbitrarily close matrices of maximal rank, that is, for every $\varepsilon>0$ there exists an $m \times n$ matrix $B$ with rank $B=$ nsuch that the Euclidean matrix norm $\|A-B\|<\varepsilon$.

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Problem 21

True or false: (a) If $\lambda$ is an eigenvalue of both $A$ and $B$, then it is an eigenvalue of the sum $A+B$. (b) If $\mathrm{v}$ is an eigenvector of both $A$ and $B$, then it is an eigenvector of $A+B$.

Victor Salazar
Victor Salazar
Numerade Educator
02:26

Problem 21

Prove that two complete matrices $A, B$ have the same eigenvalues (with multiplicities) if and only if they are similar, i.e., $B=S^{-1} A S$ for some nonsingular matrix $S$.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:06

Problem 21

An elastic body has stress tensor $T=\left(\begin{array}{lll}3 & 1 & 2 \\ 1 & 3 & 1 \\ 2 & 1 & 3\end{array}\right)$. Find the principal stretches and principal directions of stretch.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:28

Problem 21

Let $J$ be a Jordan matrix. (a) Prove that $J^k$ is a complete matrix for some $k \geq 1$ if and only if either $J$ is diagonal, or $J$ is nilpotent with $J^k=0$. (b) Suppose that $A$ is an incomplete matrix such that $A^k$ is complete for some $k \geq 2$. Prove that $A^k=0$, and hence $A$ is nilpotent. (A simpler version of this problem appears in Exercise 8.3.8.)

Nick Johnson
Nick Johnson
Numerade Educator
00:40

Problem 21

True or false: If $\operatorname{det} A>1$, then $A$ is not ill-conditioned.

Taylor Shimono
Taylor Shimono
Numerade Educator
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Problem 22

True or false: If $\lambda$ is an eigenvalue of $A$ and $\mu$ is an eigenvalue of $B$, then $\lambda \mu$ is an eigenvalue of the matrix product $C=A B$.

Victor Salazar
Victor Salazar
Numerade Educator
05:35

Problem 22

Let $B$ be obtained from $A$ by permuting both its rows and columns using the same permutation $\pi$, so $b_{i j}=a_{\pi(i), \pi(j)}$. Prove that $A$ and $B$ have the same eigenvalues. How are their eigenvectors related?

Jacob Fry
Jacob Fry
Numerade Educator

Problem 22

Given a solid body spinning around its center of mass, the eigenvectors of its positive definite inertia tensor prescribe three mutually orthogonal principal directions of rotation, while the corresponding eigenvalues are the moments of inertia. Given the inertia tensor $T=\left(\begin{array}{lll}2 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 1 & 2\end{array}\right)$, find the principal directions and moments of inertia.

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04:01

Problem 22

Cayley-Hamilton Theorem: Let $p_A(\lambda)=\operatorname{det}(A-\lambda I)$ be the characteristic polynomial of $A$. (a) Prove that if $D$ is a diagonal matrix, $\operatorname{then}^{\dagger} p_D(D)=0$.
Hint: Leave $p_D(\lambda)$ in factored form. (b) Prove that if $A$ is complete, then $p_A(A)=0$. (c) Prove that if $J$ is a Jordan block, then $p_J(J)=\mathrm{O}$. (d) Prove that this also holds if $J$ is a Jordan matrix. (e) Prove the Cayley-Hamilton Theorem: a square matrix $A$ satisfies its own characteristic equation: $p_A(A)=\mathrm{O}$.

Sanchit Jain
Sanchit Jain
Numerade Educator
02:34

Problem 22

Let $A=\left(\begin{array}{rrr}6 & -4 & 1 \\ -4 & 6 & -1 \\ 1 & -1 & 11\end{array}\right)$, and let $E=\{\mathbf{y}=A \mathbf{x} \mid\|\mathbf{x}\|=1\}$ be the image of the unit Euclidean sphere under the linear map induced by $A$. (a) Explain why $E$ is an ellipsoid and write down its equation. (b) What are its principal axes and their lengths - the semiaxes of the ellipsoid? (c) What is the volume of the solid ellipsoidal domain enclosed by $E$ ?

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 23

Let $A$ and $B$ be $n \times n$ matrices. Prove that the matrix products $A B$ and $B A$ have the same eigenvalues. Hint: How should the eigenvectors be related?

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 23

Truc or false: If $A$ is a complete upper triangular matrix, then it has an upper triangular eigenvector matrix $S$.

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 23

Let $K$ be a positive definite $2 \times 2$ matrix. (a) Explain why the quadratic equation $\mathbf{x}^T K \mathbf{x}=1$ defines an ellipse. Prove that its principal axes are the eigenvectors of $K$, and the semi-axes are the reciprocals of the square roots of the eigenvalues.
(b) Graph and describe the following curves:
(i) $x^2+4 y^2=1$,
(ii) $x^2+x y+y^2=1$,
(iii) $3 x^2+2 x y+y^2=1$.
(c) What sort of curve(s) does $\mathbf{x}^T K \mathbf{x}=1$ describe if $K$ is not positive definite?

Victor Salazar
Victor Salazar
Numerade Educator

Problem 23

Minimal polynomial: Let $A$ be an $n \times n$ matrix. By definition, the minimal polynomial of $A$ is the monic polynomial $m_A(t)=t^k+c_{k-1} t^{k-1}+\cdots+c_1 t+c_0$ of minimal degree $k$ that annihilates $A$, so $m_A(A)=A^k+c_{k-1} A^{k-1}+\cdots+c_1 A+c_0 \mathrm{I}=0$. (a) Prove that the monic minimal polynomial $m_A$ is unique. (b) (c) Prove that if $r(t)$ is any other polynomial such that $r(A)=0$, then $r(t)=q(t) m_A(t)$ for some polynomial $q(t)$. (d) Prove that the matrix's minimal polynomial is a factor of its characteristic polynomial, so $p_A(t)=q_A(t) m_A(t)$ for some polynomial $q_A(t)$. Hint: Use the Cayley-Hamilton Theorem in Exercise 8.6.22. (e) Prove that if $A$ has all distinct eigenvalues, then $p_A=m_A$. (f) Prove that $p_A=m_A$ if and only if no two Jordan blocks have the same eigenvalue.

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02:00

Problem 23

Let $A$ be a nonsingular $2 \times 2$ matrix with singular value decomposition $A=P \Sigma Q^T$ and singular values $\sigma_1 \geq \sigma_2>0$. (a) Prove that the image of the unit (Euclidean) circle under the linear transformation defined by $A$ is an ellipse, $E=\{A \mathbf{x} \mid\|\mathbf{x}\|=1\}$, whose principal axes are the columns $\mathbf{p}_1, \mathbf{p}_2$ of $P$, and whose corresponding semi-axes are the singular values $\sigma_1, \sigma_2$. (b) Show that if $A$ is symmetric, then the ellipse's principal axes are the eigenvectors of $A$ and the semi-axes are the absolute values of its eigenvalues. (c) Prove that the area of $E$ equals $\pi|\operatorname{det} A|$. (d) Find the principal axes, semi-axes, and area of the ellipses defined by $A$ is singular?
(i) $\left(\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right)$,
(ii) $\left(\begin{array}{rr}2 & 1 \\ -1 & 2\end{array}\right)$,
(iii) $\left(\begin{array}{ll}5 & -4 \\ 0 & -3\end{array}\right)$.
(e) What happens if

Victor Salazar
Victor Salazar
Numerade Educator
02:00

Problem 24

(a) Prove that if $\lambda \neq 0$ is a nonzero eigenvalue of the nonsingular matrix $A$, then $1 / \lambda$ is an eigenvalue of $A^{-1}$. (b) What happens if $A$ has 0 as an eigenvalue?

Sanchit Jain
Sanchit Jain
Numerade Educator
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Problem 24

How many different diagonal forms does an $n \times n$ diagonalizable matrix have?

Nick Johnson
Nick Johnson
Numerade Educator
02:58

Problem 24

Let $K$ be a positive definite $3 \times 3$ matrix. (a) Prove that the quadratic equation $\mathbf{x}^T K \mathbf{x}=1$ defines an ellipsoid in $\mathbb{R}^3$. What are its principal axes and semi-axes?
(b) Describe the surface defined by the equation $11 x^2-8 x y+20 y^2-10 x z+8 y z+11 z^2=1$.

James Kiss
James Kiss
Numerade Educator

Problem 24

Prove Lemma 8.54.

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02:16

Problem 24

Optimization Principles for Singular Values: Let $A$ be any nonzero $m \times n$ matrix. Prove that (a) $\sigma_1=\max \{\|A \mathbf{u}\| \mid\|\mathbf{u}\|=1\}$. (b) Is the minimum the smallest singular value? (c) Can you design optimization principles for the intermediate singular values?

Victor Salazar
Victor Salazar
Numerade Educator
02:26

Problem 25

(a) Prove that if $|\operatorname{det} A|>1$, then $A$ has at least one eigenvalue with $|\lambda|>1$.
(b) If $\mid$ det $A \mid<1$, are all eigenvalues $|\lambda|<1$ ? Prove or find a counterexample.

Victor Salazar
Victor Salazar
Numerade Educator
03:43

Problem 25

Characterize all complete matrices that are their own inverses: $A^{-1}=A$. Write down a non-diagonal example.

Pam Russell
Pam Russell
Numerade Educator
11:54

Problem 25

Prove that $A=A^T$ has a repeated eigenvalue if and only if it commutes, $A J=J A$, with a nonzero skew-symmetric matrix: $J^T=-J \neq 0$.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
07:23

Problem 25

Prove that the $n$ vectors constructed in the proof of Theorem 8.51 are linearly independent and hence form a Jordan basis.
Hint: Suppose that some linear combination vanishes. Apply $B$ to the equation, and then use the fact that we started with a Jordan basis for $W=\operatorname{img} B$.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:06

Problem 25

Let $A$ be a square matrix. Prove that its maximal eigenvalue is smaller than its maximal singular value: $\max \left|\lambda_i\right| \leq \max \sigma_i$. Hint: Use Exercise 8.7.24.

Nick Johnson
Nick Johnson
Numerade Educator
00:38

Problem 26

Prove that $A$ is a singular matrix if and only if 0 is an eigenvalue.

Nicholas Barvinok
Nicholas Barvinok
Numerade Educator

Problem 26

Two $n \times n$ matrices $A, B$ are said to be simultaneously diagonalizable if there is a nonsingular matrix $S$ such that both $S^{-1} A S$ and $S^{-1} B S$ are diagonal matrices.
(a) Show that simultaneously diagonalizable matrices commute: $A B=B A$.
(b) Prove that the converse is valid, provided that one of the matrices has no multiple eigenvalues. (c) Is every pair of commuting matrices simultaneously diagonalizable?

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Problem 26

Find all positive definite orthogonal matrices.

Nick Johnson
Nick Johnson
Numerade Educator
02:05

Problem 26

Find all invariant subspaces of the following matrices: (a) $\left(\begin{array}{rr}4 & 0 \\ -3 & 4\end{array}\right),(b)\left(\begin{array}{rr}-1 & -1 \\ 4 & -5\end{array}\right)$,
(c) $\left(\begin{array}{rrr}0 & -1 & 1 \\ 1 & 1 & -1 \\ 3 & 3 & -4\end{array}\right)$.
(d) $\left(\begin{array}{lll}3 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 3\end{array}\right)$,
(e) $\left(\begin{array}{llll}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1\end{array}\right)$,
(f)
$\left(\begin{array}{rrrr}-1 & 0 & 1 & 2 \\ 0 & 1 & 0 & 1 \\ -1 & -4 & 1 & -2 \\ 0 & 1 & 0 & 1\end{array}\right)$

Victor Salazar
Victor Salazar
Numerade Educator
03:13

Problem 26

Compute the Euclidean matrix norm of the following matrices.
(a) $\left(\begin{array}{ll}\frac{1}{2} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{6}\end{array}\right)$,
(b) $\left(\begin{array}{rr}\frac{5}{3} & \frac{4}{3} \\ -\frac{7}{6} & -\frac{5}{6}\end{array}\right)$,
(c) $\left(\begin{array}{rr}\frac{2}{7} & -\frac{2}{7} \\ -\frac{2}{7} & \frac{6}{7}\end{array}\right)$,
(d) $\left(\begin{array}{rr}\frac{1}{4} & \frac{3}{2} \\ -\frac{1}{2} & \frac{5}{4}\end{array}\right)$,
(e) $\left(\begin{array}{rrr}\frac{2}{7} & \frac{2}{7} & -\frac{4}{7} \\ 0 & \frac{2}{7} & \frac{6}{7} \\ \frac{2}{7} & \frac{4}{7} & \frac{2}{7}\end{array}\right)$,
(f) $\left(\begin{array}{rrr}0 & .1 & .8 \\ -.1 & 0 & .1 \\ -.8 & -.1 & 0\end{array}\right)$,
(g) $\left(\begin{array}{rrr}1 & -\frac{2}{3} & -\frac{2}{3} \\ 1 & -\frac{1}{3} & -1 \\ \frac{1}{3} & -\frac{2}{3} & 0\end{array}\right)$,
(h) $\left(\begin{array}{rrr}\frac{1}{3} & 0 & 0 \\ -\frac{1}{3} & 0 & \frac{1}{3} \\ 0 & \frac{2}{3} & \frac{1}{3}\end{array}\right)$.

Vishnu P
Vishnu P
Numerade Educator
05:10

Problem 27

Prove that every nonzero vector $\mathbf{0} \neq \mathrm{v} \in \mathbb{R}^n$ is an eigenvector of $A$ if and only if $A$ is a scalar multiple of the identity matrix.

Gennady Notowidigdo
Gennady Notowidigdo
Numerade Educator
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Problem 27

(a) Prove that every positive definite matrix $K$ has a unique positive definite square root, i.e., a matrix $B>0$ satisfying $B^2=K$.
(b) Find the positive definite square roots of the following matrices:
(i) $\left(\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right)$
(ii) $\left(\begin{array}{rr}3 & -1 \\ -1 & 1\end{array}\right)$
(iii) $\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 9\end{array}\right)$,
(iv) $\left(\begin{array}{rrr}6 & -4 & 1 \\ -4 & 6 & -1 \\ 1 & -1 & 11\end{array}\right)$.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 27

The method for constructing a Jordan basis in Example 8.52 is simplified due to the fact that each eigenvalue admits only one Jordan block. On the other hand, the method used in the proof of Theorem 8.51 is rather impractical Devise a general method for constructing a Jordan basis for an arbitrary matrix, paying careful attention to how the Jordan chains having the same eigenvalue are found.

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02:32

Problem 27

Find a matrix $A$ whose Euclidean matrix norm satisfies $\left\|A^2\right\| \neq\|A\|^2$.

Manisha Sarker
Manisha Sarker
Numerade Educator
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Problem 28

How many unit (norm 1) eigenvectors correspond to a given eigenvalue of a matrix?

Michelle Z.
Michelle Z.
Numerade Educator
06:04

Problem 28

The Polar Decomposition: Prove that every invertible matrix $A$ has a polar decomposition, written $A=Q B$, into the product of an orthogonal matrix $Q$ and a positive definite matrix $B>0$. Show that if $\operatorname{det} A>0$, then $Q$ is a proper orthogonal matrix. Hint: Look at the Gram matrix $K=A^T A$ and use Exercise 8.5.27.
Remark. In mechanics, if $A$ represents the deformation of a body, then $Q$ represents a rotation, while $B$ represents a stretching along the orthogonal eigendirections of $K$. Thus, every linear deformation of an elastic body can be decomposed into a pure stretching transformation followed by a rotation.

Jack Chen
Jack Chen
Numerade Educator

Problem 28

Let $K>0$ be a positive definite matrix. Characterize the matrix norm induced by the inner product $\langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^T K \mathbf{y}$. Hint: Use Exercise 8.5.45.

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Problem 29

True or false: (a) Performing an elementary row operation of type \#1 does not change the eigenvalues of a matrix. (b) Interchanging two rows of a matrix changes the sign of its eigenvalues. (c) Multiplying one row of a matrix by a scalar multiplies one of its eigenvalues by the same scalar.

Donna Densmore
Donna Densmore
Numerade Educator
04:12

Problem 29

Find the polar decompositions $A=Q B$, as defined in Exercise 8.5.28, of the following matrices:
(a) $\left(\begin{array}{ll}0 & 1 \\ 2 & 0\end{array}\right)$,
(b) $\left(\begin{array}{rr}2 & -3 \\ 1 & 6\end{array}\right)$,
(c) $\left(\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}0 & -3 & 8 \\ 1 & 0 & 0 \\ 0 & 4 & 6\end{array}\right)$,
(e) $\left(\begin{array}{rrr}1 & 0 & 1 \\ 1 & -2 & 0 \\ 1 & 1 & 0\end{array}\right)$.

Supratim Pal
Supratim Pal
Numerade Educator

Problem 29

Let $A=\left(\begin{array}{rr}1 & 1 \\ 1 & -2\end{array}\right)$. Compute the matrix norm $\|A\|$ using the following norms in $\mathbb{R}^2$ :
(a) the weighted $\infty$ norm $\|\mathbf{v}\|=\max \left\{2\left|v_1\right|, 3\left|v_2\right|\right\}$; (b) the weighted 1 norm
$\|\mathbf{v}\|=2\left|v_1\right|+3\left|v_2\right| ;(c)$ the weighted inner product norm $|\mathbf{v}|=\sqrt{2 v_1^2+3 v_2^2}$;
(d) the norm associated with the positive definite matrix $K=\left(\begin{array}{rr}2 & -1 \\ -1 & 2\end{array}\right)$.

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03:02

Problem 30

(a) True or false: If $\lambda_1, \mathrm{v}_1$ and $\lambda_2, \mathrm{v}_2$ solve the eigenvalue equation (8.12) for a given matrix $A$, so does $\lambda_1+\lambda_2, v_1+\mathbf{v}_2$. (b) Explain what this has to do with linearity.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
07:51

Problem 30

The Spectral Theorem for Hermitian Matrices. Prove that a complex Hermitian matrix can be factored as $H=U \Lambda U^{\dagger}$ where $U$ is a unitary matrix and $\Lambda$ is a real diagonal matrix. Hint: See Exercises 4.3.25, 8.5.7.

Sam Stansfield
Sam Stansfield
Numerade Educator

Problem 30

Let $A$ be an $n \times n$ matrix with singular value vector $\sigma=\left(\sigma_1, \ldots, \sigma_r\right)$. Prove that
(a) $\|\boldsymbol{\sigma}\|_{\infty}=\|A\|_2 ;$ (b) $\|\sigma\|_2=\|A\|_F$, the Frobenius norm of Exercise 3.3.51.
Remark. The 1 norm of the singular value vector $\|\boldsymbol{\sigma}\|_1$ also defines a useful matrix norm, known as the $K y$ Fan norm.

$$
\sum_{i=1}^r \sigma_i^2=\sum_{i=1}^m \sum_{j=1}^n a_{i j}^2
$$
8.7.32. Let $A$ be a nonsingular square matrix. Prove the following formulas for its condition number:
(a) $\kappa(A)=\frac{\max \{\|A \mathbf{u}\| \mid\|\mathbf{u}\|=1\}}{\min \{\|A \mathbf{u}\| \mid\|\mathbf{u}\|=1\}}$,
(b) $\kappa(A)=\|A\|_2\left\|A^{-1}\right\|_2$.
8.7.33. Find the pseudoinverse of the following matrices:
(a) $\left(\begin{array}{rr}1 & -1 \\ -3 & 3\end{array}\right)$
(b) $\left(\begin{array}{rr}1 & -2 \\ 2 & 1\end{array}\right)$,
(c) $\left(\begin{array}{rr}2 & 0 \\ 0 & -1 \\ 0 & 0\end{array}\right)$,
(d) $\left(\begin{array}{rrr}0 & 0 & 1 \\ 0 & -1 & 0 \\ 0 & 0 & 0\end{array}\right)$,
(e) $\left(\begin{array}{rrr}1 & -1 & 1 \\ -2 & 2 & -2\end{array}\right)$,
(f) $\left(\begin{array}{ll}1 & 3 \\ 2 & 6 \\ 3 & 9\end{array}\right)$,
(g) $\left(\begin{array}{rrr}1 & 2 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & -1\end{array}\right)$.

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Problem 31

As in (4.35), an elementary reflection matrix has the form $Q=\mathrm{I}-2 \mathbf{u u}^T$, where $\mathbf{u} \in \mathbb{R}^n$ is a unit vector. (a) Find the eigenvalues and eigenvectors of the elementary
reflection matrices for the unit vectors $(i)\left(\begin{array}{l}1 \\ 0\end{array}\right)$,
(ii) $\left(\begin{array}{c}\frac{3}{5} \\ \frac{1}{5}\end{array}\right)$,
(iii) $\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)$,
$\left(\begin{array}{c}\frac{1}{\sqrt{2}} \\ 0 \\ -\frac{1}{\sqrt{2}}\end{array}\right)$
(b) What are the eigenvalues and eigenvectors of a general elementary reflection matrix?

Victor Salazar
Victor Salazar
Numerade Educator
01:52

Problem 31

Find the spectral factorization, as in Exercise 8.5.30, of the following Hermitian matrices:
(a) $\left(\begin{array}{cc}3 & 2 \mathrm{i} \\ -2 \mathrm{i} & 6\end{array}\right)$,
(b) $\left(\begin{array}{cc}6 & 1-2 \mathrm{i} \\ 1+2 \mathrm{i} & 2\end{array}\right)$,
(c) $\left(\begin{array}{ccc}-1 & 5 i & -4 \\ -5 i & -1 & 4 i \\ -4 & -4 i & 8\end{array}\right)$.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 31

Let $A$ be an $m \times n$ matrix with singular values $\sigma_1, \ldots, \sigma_r$. Prove that

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03:07

Problem 32

Let $A$ and $B$ be similar matrices, so $B=S^{-1} A S$ for some nonsingular matrix $S$.
(a) Prove that $A$ and $B$ have the same characteristic polynomial: $p_B(\lambda)=p_A(\lambda)$.
(b) Explain why similar matrices have the same eigenvalues. (c) Do they have the same eigenvectors? If not, how are their eigenvectors related? (d) Prove that the converse to part (c) is false by showing that $\left(\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right)$ and $\left(\begin{array}{rr}1 & 1 \\ -1 & 3\end{array}\right)$ have the same eigenvalues, but are not similar.

Anthony Ramos
Anthony Ramos
Numerade Educator
06:12

Problem 32

The Spectral Decomposition: Let $A$ be a symmetric matrix with distinct eigenvalues $\lambda_1, \ldots, \lambda_k$. Let $V_j=\operatorname{ker}\left(A-\lambda_j \mathrm{I}\right)$ denote the eigenspace corresponding to $\lambda_j$, and let $P_j$ be the orthogonal projection matrix onto $V_j$, as defined in Exercise 4.4.9. (i) Prove that the spectral factorization (8.35) can be rewritten as
$$
A=\lambda_1 P_1+\lambda_2 P_2+\cdots+\lambda_k P_k \text {. }
$$
expressing $A$ as a linear combination of projection matrices. (ii) Write out the spectral decomposition (8.37) for the matrioes in Exercise 8.5.13. (iii) Show that
$$
\mathrm{I}=P_1+P_2+\cdots+P_k, \quad \text { while } \quad P_i^2=P_i, \quad P_i P_j=0 \text { for } i \neq j .
$$
(iv) Show that if $p(t)$ is any polynomial, then
$$
p(A)=p\left(\lambda_1\right) P_1+p\left(\lambda_2\right) P_2+\cdots+p\left(\lambda_k\right) P_k .
$$

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
01:44

Problem 32

Let $A$ be a nonsingular square matrix. Prove the following formulas for its condition number:
(a) $\kappa(A)=\frac{\max \{\|A \mathbf{u}\| \mid\|\mathbf{u}\|=1\}}{\min \{\|A \mathbf{u}\| \mid\|\mathbf{u}\|=1\}}$,
(b) $\kappa(A)=\|A\|_2\left\|A^{-1}\right\|_2$.

Patrick Burns
Patrick Burns
Numerade Educator
07:43

Problem 33

Let $A$ be a nonsingular $n \times n$ matrix with characteristic polynomial $p_A(\lambda)$.
(a) Explain how to construct the characteristic polynomial $p_{A^{-1}}(\lambda)$ of its inverse directly from $p_A(\lambda)$
(b) Check your result when $A=(i)\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right)$,
(ii) $\left(\begin{array}{rrr}1 & 4 & 4 \\ -2 & -1 & 0 \\ 0 & 2 & 3\end{array}\right)$

(The converse is also true; see Exercise 8.6.20.)

Anthony Ramos
Anthony Ramos
Numerade Educator
01:36

Problem 33

Find the minimum and maximum values of the quadratic form $5 x^2+4 x y+5 y^2$ where $x, y$ are subject to the constraint $x^2+y^2=1$.

Aymara Gallardo
Aymara Gallardo
Numerade Educator
13:14

Problem 33

Find the psendoinverse of the following matrices:
(a) $\left(\begin{array}{rr}1 & -1 \\ -3 & 3\end{array}\right)$,
(b) $\left(\begin{array}{rr}1 & -2 \\ 2 & 1\end{array}\right)$,
(c) $\left(\begin{array}{rr}2 & 0 \\ 0 & -1 \\ 0 & 0\end{array}\right)$,
(d) $\left(\begin{array}{rrr}0 & 0 & 1 \\ 0 & -1 & 0 \\ 0 & 0 & 0\end{array}\right)$,
(e) $\left(\begin{array}{rrr}1 & -1 & 1 \\ -2 & 2 & -2\end{array}\right)$,
(f) $\left(\begin{array}{ll}1 & 3 \\ 2 & 6 \\ 3 & 9\end{array}\right)$,
(g) $\left(\begin{array}{rrr}1 & 2 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & -1\end{array}\right)$.

Carole Wastog
Carole Wastog
Numerade Educator
05:10

Problem 34

Prove that the only eigenvalue of a nilpotent matrix, cf. Exercise 1.3.12, is 0.

Gennady Notowidigdo
Gennady Notowidigdo
Numerade Educator
02:54

Problem 34

Find the minimum and maximum values of the quadratic form
$$
2 x^2+x y+2 x z+2 y^2+2 z^2 \text { where } x, y, z \text { are required to satisfy } x^2+y^2+z^2=1 \text {. }
$$

Lucas Finney
Lucas Finney
Numerade Educator
00:46

Problem 34

Use the pseudoinverse to find the least squares solution of minimal norm to the following linear systems:
(a)
$$
\begin{aligned}
x+y & =1, \\
3 x+3 y & =-2 ;
\end{aligned}
$$
(b)
$$
\begin{array}{rlrl}
x+y+z & =5, & x-3 y & =2, \\
2 x-y+z & =2 ; & \text { (c) } 2 x+y & =-1, \\
x+y & =0 .
\end{array}
$$
(c) $2 x+y=-1$,

Victor Salazar
Victor Salazar
Numerade Educator
11:41

Problem 35

Given an idempotent matrix, so that $P=P^2$, find all its eigenvalues and eigenvectors.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
01:32

Problem 35

What is the minimum and maximum values of the following rational functions:
(a) $\frac{3 x^2-2 y^2}{x^2+y^2}$,
(b) $\frac{x^2-3 x y+y^2}{x^2+y^2}$,
(c) $\frac{3 x^2+x y+5 y^2}{x^2+y^2}$,
(d) $\frac{2 x^2+x y+3 x z+2 y^2+2 z^2}{x^2+y^2+z^2}$

Donald Albin
Donald Albin
Numerade Educator
02:03

Problem 35

Prove that the pseudoinverse satisfies the following identities: (a) $\left(A^{+}\right)^{+}=A$,
(b) $A A^{+} A=A$,
(c) $A^{+} A A^{+}=A^{+}$,
(d) $\left(A A^{+}\right)^T=A A^{+}$,
(e) $\left(A^{+} A\right)^T=A^{+} A$.

Griffin Goodwin
Griffin Goodwin
Numerade Educator
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Problem 36

(a) Prove that every real $3 \times 3$ matrix has at least one real eigenvalue.
(b) Find a real $4 \times 4$ matrix with no real eigenvalues.
(c) Can you find a real $5 \times 5$ matrix with no real eigenvalues?

Victor Salazar
Victor Salazar
Numerade Educator
02:24

Problem 36

Find the minimum and maximum values of $q(\mathbf{x})=\sum_{i=1}^{n-1} x_i x_{i+1}$ for $|\mathbf{x}|^2=1$. Hint: See Exercise 8.2.47.

Jack Chen
Jack Chen
Numerade Educator
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Problem 36

Suppose $\mathbf{b} \in \operatorname{img} A$ and $\operatorname{ker} A=\{0\}$. Prove that $\mathbf{x}^*=A^{+} \mathbf{b}$ is the unique solution to the linear system $A \mathrm{x}=\mathrm{b}$. What if $\operatorname{ker} A \neq\{0\}$ ?

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 37

(a) Show that if $A$ is a matrix such that $A^4=1$, then the only possible eigenvalues of $A$ are $1,-1, i$, and $-\mathrm{i}$.
(b) Give an example of a real matrix that has all four numbers as eigenvalues.

Michelle Z.
Michelle Z.
Numerade Educator
19:57

Problem 37

Write down and solve an optimization principle characterizing the largest and smallest eigenvalues of the following positive definite matrices:
(a) $\left(\begin{array}{rr}2 & -1 \\ -1 & 3\end{array}\right)$,
(b) $\left(\begin{array}{ll}4 & 1 \\ 1 & 4\end{array}\right)$.
(c) $\left(\begin{array}{rrr}6 & -4 & 1 \\ -4 & 6 & -1 \\ 1 & -1 & 11\end{array}\right)$,
(d) $\left(\begin{array}{rrr}4 & -1 & -2 \\ -1 & 4 & -1 \\ -2 & -1 & 4\end{array}\right)$.

Chris Trentman
Chris Trentman
Numerade Educator

Problem 37

Choose a direction for each of the edges and write down the incidence matrix $A$ for the graph sketched in Figure 8.4. Verify that its graph Laplacian $(8.62)$ equals $K=A^T A$.

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01:49

Problem 38

(a) Prove that if $\lambda$ is an eigenvalue of $A$, then $\lambda^n$ is an eigenvalue of $A^n$.
(b) State and prove a converse.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
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Problem 38

Write down a maximization principle that characterizes the middle eigenvalue of the matrices in parts $(c)$ and $(d)$ of Exercise 8.5.37.

Victor Salazar
Victor Salazar
Numerade Educator
00:19

Problem 38

Determine the spectrum for the graphs in Exercise 2.6.3.

Nidhi Singhi
Nidhi Singhi
Numerade Educator
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Problem 39

True or false: All the eigenvalues of an $n \times n$ permutation matrix are real.

Victor Salazar
Victor Salazar
Numerade Educator
03:39

Problem 39

Given a $3 \times 3$ symmetric matrix, formulate two distinct ways of characterizing its middle eigenvalue $\lambda_2$.

Nikhil Kumar Rajpurohit
Nikhil Kumar Rajpurohit
Numerade Educator

Problem 39

Determine the spectrum of a graph given by the edges of (i) a triangle; (ii) a square; (iii) a pentagon. Can you determine the formula for the spectrum of the graph given by an $n$ sided polygon? Hint: See Exercise 8.2.49.

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02:24

Problem 40

(a) Show that if all the row sums of $A$ are equal to 1 , then $A$ has 1 as an eigenvalue.
(b) Suppose all the column sums of $A$ are equal to 1 . Does the same result hold?

Sanchit Jain
Sanchit Jain
Numerade Educator
00:29

Problem 40

Suppose $K>0$. What is the maximum value of $q(\mathbf{x})=\mathbf{x}^T K \mathbf{x}$ when $\mathbf{x}$ is constrained to a sphere of radius $\|\mathbf{x}\|=r$ ?

Yujie Wang
Yujie Wang
College of San Mateo
04:05

Problem 40

Determine the spectrum for the trees in Exercise 2.6.9. Can you make any conjectures about the nature of the spectrum of a graph that is a tree?

Chris Trentman
Chris Trentman
Numerade Educator
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Problem 41

Prove that if $\mathrm{v}$ is an eigenvector of $A$ with eigenvalue $\lambda$ and $\mathrm{w}$ is an eigenvector of $A^T$ with a different eigenvalue $\mu \neq \lambda$, then $\mathbf{v}$ and $\mathbf{w}$ are orthogonal vectors with respect to the dot product. Illustrate this result when $(i) A=\left(\begin{array}{rr}0 & -1 \\ 2 & 3\end{array}\right)$,
(ii) $A=\left(\begin{array}{rrr}5 & -4 & 2 \\ 5 & -4 & 1 \\ -2 & 2 & -3\end{array}\right)$.

Victor Salazar
Victor Salazar
Numerade Educator
00:18

Problem 41

Let $K>0$. Prove the product formula
$$
\max \left\{\mathbf{x}^T K \mathbf{x} \mid\|\mathbf{x}\|=1\right\} \min \left\{\mathbf{x}^T K^{-1} \mathbf{x} \mid\|\mathbf{x}\|=1\right\}=1 .
$$

Linh Vu
Linh Vu
Numerade Educator
03:45

Problem 42

Let $Q$ be an orthogonal matrix. (a) Prove that if $\lambda$ is an eigenvalue, then so is $1 / \lambda$.
(b) Prove that all its eigenvalues are complex numbers of modulus $|\lambda|=1$. In particular, the only possible real eigenvalues of an orthogonal matrix are \pm 1 . (c) Suppose $\mathbf{v}=\mathbf{x}+i \mathbf{y}$ is a complex eigenvector corresponding to a non-real eigenvalue. Prove that its real and imaginary parts are orthogonal vectors having the same Euclidean norm.

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator

Problem 42

Write out the details in the proof of Theorem 8.42 .

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07:15

Problem 43

(a) Prove that every $3 \times 3$ proper orthogonal matrix has +1 as an eigenvalue.
(b) True or false: An improper $3 \times 3$ orthogonal matrix has -1 as an eigenvalue.

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator
01:12

Problem 43

Reformulate Theorem 8.42 as a minimum principle for intermediate eigenvalues.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:01

Problem 44

(a) Show that the linear transformation defined by a $3 \times 3$ proper orthogonal matrix corresponds to rotating through an angle around a line through the origin in $\mathbb{R}^3$ - the axis of the rotation. Hint: Use Exercise $8.2 .43(a)$.
(b) Find the axis and angle of rotation of the orthogonal matrix
$$
\left(\begin{array}{rrr}
\frac{3}{5} & 0 & \frac{4}{5} \\
-\frac{4}{13} & \frac{12}{13} & \frac{3}{13} \\
-\frac{48}{65} & -\frac{5}{13} & \frac{36}{65}
\end{array}\right)
$$

Raj Bala
Raj Bala
Numerade Educator

Problem 44

Under the set-up of Theorem 8.42, explain why
$$
\lambda_j=\max \left\{\frac{\mathbf{v}^T K \mathbf{v}}{|\mathbf{v}|^2} \mid \mathbf{v} \neq \mathbf{0}, \quad \mathbf{v} \cdot \mathbf{v}_1=\cdots=\mathbf{v} \cdot \mathbf{v}_{j-1}=0\right\} .
$$

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Problem 45

Prove that every proper affine isometry $F(\mathbf{x})=Q \mathbf{x}+\mathbf{b}$ of $\mathbb{R}^3$, where det $Q=+1$, is one of the following: (a) a translation $\mathbf{x}+\mathbf{b},(b)$ a rotation centered at some point of $\mathbb{R}^3$, or (c) a screw motion consisting of a rotation around an axis followed by a translation in the direction of the axis. Hint: Use Exercise 8.2.44.

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02:06

Problem 45

(a) Let $K, M$ be positive definite $n \times n$ matrices and $\lambda_1 \geq \cdots \geq \lambda_n$ be their generalized eigenvalues, as in Exercise 8.5.9. Prove that that the largest generalized eigenvalue can be characterized by the maximum principle $\lambda_1=\max \left\{\mathbf{x}^T K \mathbf{x} \mid \mathbf{x}^T M \mathbf{x}=1\right\}$.
Hint: Use Exercise 8.5.27.
(b) Prove the alternative maximum principle $\lambda_1=$ $\max \left\{\frac{\mathrm{x}^T K \mathrm{x}}{\mathrm{x}^T M \mathrm{x}} \mid \mathrm{x} \neq \mathbf{0}\right\}$.
(c) How would you characterize the smallest generalized eigenvalue? (d) An intermediate generalized eigenvalue?

Nick Johnson
Nick Johnson
Numerade Educator

Problem 46

Suppose $Q$ is an orthogonal matrix. (a) Prove that $K=2 \mathrm{I}-Q-Q^T$ is a positive semi-definite matrix. (b) Under what conditions is $K>0$ ?

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Problem 46

Use Exercise 8.5.45 to find the minimum and maximum of the rational functions
(a) $\frac{3 x^2+2 y^2}{4 x^2+5 y^2}$,
(b) $\frac{x^2-x y+2 y^2}{2 x^2-x y+y^2}$
(c) $\frac{2 x^2+3 y^2+z^2}{x^2+3 y^2+2 z^2}$
(d) $\frac{2 x^2+6 x y+11 y^2+6 y z+2 z^2}{x^2+2 x y+3 y^2+2 y z+z^2}$.
8.5.47. Let $A$ be a complete square matrix, not necessarily symmetric, with all positive eigenvalues. Is the associated quadratic form $q(\mathbf{x})=\mathbf{x}^T A \mathbf{x}>0$ for all $\mathbf{x} \neq \mathbf{0}$ ?

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Problem 47

Let $M_n$ be the $n \times n$ tridiagonal matrix whose diagonal entries are all equal to 0 and whose sub- and super-diagonal entries all equal 1. (a) Find the eigenvalues and eigenvectors of $M_2$ and $M_3$ directly. (b) Prove that the eigenvalues and eigenvectors of $M_n$ are explicitly given by
$$
\lambda_k=2 \cos \frac{k \pi}{n+1}, \quad \mathbf{v}_k=\left(\sin \frac{k \pi}{n+1}, \sin \frac{2 k \pi}{n+1}, \ldots \sin \frac{n k \pi}{n+1}\right)^T, \quad k=1, \ldots, n .
$$
How do you know that there are no other eigenvalues?

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15:48

Problem 48

Let $a, b \in \mathbb{R}$. Determine the eigenvalues and eigenvectors of the $n \times n$ tridiagonal matrix with all diagonal entries equal to $a$ and all sub- and super-diagonal entries equal to b. Hint: See Exercises 8.2 .19 and 8.2 .47 .

Chris Trentman
Chris Trentman
Numerade Educator
01:53

Problem 49

Find a formula for the eigenvalues of the tricirculant $n \times n$ matrix $Z_n$ that has 1's on the sub- and super-diagonals as well as its $(1, n)$ and $(n, 1)$ entries, while all other entries are 0. Hint: Use Exercise 8.2.47 as a guide.

Michelle Z.
Michelle Z.
Numerade Educator

Problem 50

Let $A$ be an $n \times n$ matrix with eigenvalues $\lambda_1, \ldots, \lambda_k$, and $B$ an $m \times m$ matrix with eigenvalues $\mu_1, \ldots, \mu_l$. Show that the $(m+n) \times(m+n)$ block diagonal matrix $D=\left(\begin{array}{ll}A & 0 \\ O & B\end{array}\right)$ has eigenvalues $\lambda_1, \ldots, \lambda_k, \mu_1, \ldots, \mu_l$ and no others. How are the eigenvectors related?

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03:07

Problem 51

Deflation: Suppose $A$ has eigenvalue $\lambda$ and corresponding eigenvector $\mathbf{v}$. (a) Let $\mathbf{b}$ be any vector. Prove that the matrix $B=A-\mathbf{v b}^T$ also has $\mathbf{v}$ as an eigenvector, now with eigenvalue $\lambda-\beta$, where $\beta=\mathbf{v}-\mathbf{b}$. (b) Prove that if $\mu \neq \lambda-\beta$ is any other eigenvalue of $A$, then it is also an eigenvalue of $B$. Hint: Look for an eigenvector of the form $\mathbf{w}+c \mathbf{v}$, where w is an eigenvector of $A$. (c) Given a nonsingular matrix $A$ with eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$ and $\lambda_1 \neq \lambda_j$ for all $j \geq 2$, explain how to construct a deflated matrix $B$ whose eigenvalues are $0, \lambda_2, \ldots, \lambda_n$.
(d) Try out your method on the matrices $\left(\begin{array}{ll}3 & 3 \\ 1 & 5\end{array}\right)$ and $\left(\begin{array}{rrr}3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3\end{array}\right)$.

Anthony Ramos
Anthony Ramos
Numerade Educator
16:01

Problem 52

Let $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ be a $2 \times 2$ matrix. (a) Prove that $A$ satisfies its own characteristic equation, meaning $p_A(A)=A^2-(\operatorname{tr} A) A+(\operatorname{det} A) \mathrm{I}=0 . \quad$ Remark. This result is a special case of the Cayley-Hamilton Theorem, to be developed in Exercise 8.6.22.
(b) Prove the inverse formula $A^{-1}=\frac{(\operatorname{tr} A) I-A}{\operatorname{det} A}$ when $\operatorname{det} A \neq 0$.
(c) Check the Cayley-Hamilton and inverse formulas when $A=\left(\begin{array}{rr}2 & 1 \\ -3 & 2\end{array}\right)$.

Bobby Barnes
Bobby Barnes
University of North Texas
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Problem 53

For each of the following matrices,
(i) find all Gershgorin disks; (ii) plot the Gershgorin domain in the complex plane;
(iii) compute the eigenvalues and confirm the truth of the Circle Theorem 8.16:
(a) $\left(\begin{array}{rr}1 & -2 \\ -2 & 1\end{array}\right)$,
(b) $\left(\begin{array}{cc}1 & -\frac{2}{3} \\ \frac{1}{2} & -\frac{1}{6}\end{array}\right)$.
(c) $\left(\begin{array}{rr}2 & 3 \\ -1 & 0\end{array}\right)$,
(d) $\left(\begin{array}{rrr}3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3\end{array}\right)$,
(e) $\left(\begin{array}{rrr}-1 & 1 & 1 \\ 2 & 2 & -1 \\ 0 & 3 & -4\end{array}\right)$,
(f) $\left(\begin{array}{ccc}\frac{1}{2} & 0 & 0 \\ 0 & 0 & \frac{1}{3} \\ \frac{1}{4} & \frac{1}{6} & 0\end{array}\right)$.
(g) $\left(\begin{array}{rrr}0 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & -1 & 1\end{array}\right)$,
(h) $\left(\begin{array}{llll}3 & 2 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 2 & 1\end{array}\right)$.

Nick Johnson
Nick Johnson
Numerade Educator
00:54

Problem 54

True or false: The Gershgorin domain of the transpose of a matrix $A^T$ is the same as the Gershgorin domain of the matrix $A$, that is, $D_{A^T}=D_A$.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 55

(i) Explain why the eigenvalues of $A$ must lie in its refined Gershgorin domain $D_A^*=D_{A^T} \cap D_A$. (ii) Find the refined Gershgorin domains for each of the matrices in Exercise 8.2 .53 and confirm the result in part (i).

Victor Salazar
Victor Salazar
Numerade Educator
02:03

Problem 56

True or false: (a) A positive definite matrix is strictly diagonally dominant.
(b) A strictly diagonally dominant matrix is positive definite.

Nick Johnson
Nick Johnson
Numerade Educator
03:08

Problem 57

Prove that if $K$ is symmetric, strictly diagonally dominant, and each diagonal entry is positive, then $K$ is positive definite.

ET
Ed Tam
Numerade Educator
02:22

Problem 58

(a) Write down an invertible matrix $A$ whose Gershgorin domain contains 0.
(b) Can you find an example that is also strictly diagonally dominant?

JC
Jeff Christopher
Numerade Educator