Linear Algebra in Action

Harry Dym

Chapter 4 Eigenvalues and eigenvectors - all with Video Answers

Educators

Chapter Questions

Problem 1

Show that similarity is an equivalence relation, i.e., denoting similarity by $\sim$ : (1) $A \sim A$; (2) $A \sim B \Longrightarrow B \sim A$; (3) $A \sim B$ and $B \sim C \Longrightarrow A \sim C$.

Nick Johnson

Numerade Educator

05:12

Show that if $T$ is a linear transformation from a vector space $\mathcal{V}$ over $\mathbf{F}$ into itself, then the vector spaces $\mathcal{N}_{(T-\lambda I)}$ and $\mathcal{R}_{(T-\lambda I)}$ are both invariant under $T$ for each choice of $\lambda \in \mathbb{F}$.

Eleni Katirtzoglou

Numerade Educator

Problem 3

The set $\mathcal{V}$ of polynomials $p(t)$ with complex coefficients is a vector space over $\mathbb{C}$ with respect to the natural rules of vector addition and scalar multiplication. Let $T p=p^{\prime \prime}(t)+t p^{\prime}(t)$ and $S p=p^{\prime \prime}(t)+t^2 p^{\prime}(t)$. Show that the subspace $\mathcal{U}_k$ of $\mathcal{V}$ of polynomials $p(t)=c_0+c_1 t+\cdots+c_k t^k$ of degree less than or equal to $k$ is invariant under $T$ but not under $S$. Find a nonzero polynomial $p \in \mathcal{U}_3$ and a number $\lambda \in \mathbb{C}$ such that $T p=\lambda p$.

Check back soon!

Problem 4

Show that if $T$ is a linear transformation from a vector space $\mathcal{V}$ over $\mathbb{F}$ into itself, then $T^2+5 T+6 I=(T+3 I)(T+2 I)$.

Check back soon!

02:53

Problem 5

Let $T$ be a linear transformation from a vector space $\mathcal{V}$ over $\mathbb{R}$ into itself and let $\mathcal{U}$ be a two dimensional subspace of $\mathcal{V}$ with basis $\left\{\mathbf{u}_1\right.$, $\left.\mathbf{u}_2\right\}$. Show that if $T \mathbf{u}_1=\mathbf{u}_2$ and $T \mathbf{u}_2=-\mathbf{u}_1$, then $T^2 \mathbf{u}+\mathbf{u}=\mathbf{0}$ for every vector $\mathbf{u} \in \mathcal{U}$ but that there are no one dimensional subspaces of $\mathcal{U}$ that are invariant under $T$. Why? [HINT: A one dimensional subspace of $\mathcal{U}$ is equal to $\left\{\alpha\left(c_1 \mathbf{u}_1+c_2 \mathbf{u}_2\right): \alpha \in \mathbb{R}\right\}$ for some choice of $c_1, c_2 \in \mathbb{R}$ with $\left|c_1\right|+\left|c_2\right|>0$.]

Runpeng Li

Numerade Educator

02:30

Problem 6

Show that the vectors $\mathbf{u}_1, \ldots, \mathbf{u}_k$ in a Jordan chain of length $k$ are linearly independent.
If $\lambda_1, \ldots, \lambda_k$ are distinct eigenvalues of a matrix $A \in \mathbb{F}^{n \times n}$, then:

R M

Numerade Educator

Problem 7

Verify the inclusions (4.6).

Check back soon!

02:43

Problem 8

Show that the matrices
$$
A=\left[\begin{array}{rr}
1 & -1 \\
1 & 1
\end{array}\right] \quad \text { and } \quad A=\left[\begin{array}{ll}
2 & -1 \\
3 & -1
\end{array}\right]
$$
have no real eigenvalues, i.e., $\sigma(A) \cap \mathbb{R}=\emptyset$ in both cases.

Sanchit Jain

Numerade Educator

View

Problem 9

Show that although the following upper triangular matrices
$$
\left[\begin{array}{lll}
2 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 2
\end{array}\right],\left[\begin{array}{lll}
2 & 1 & 0 \\
0 & 2 & 0 \\
0 & 0 & 2
\end{array}\right],\left[\begin{array}{lll}
2 & 1 & 0 \\
0 & 2 & 1 \\
0 & 0 & 2
\end{array}\right]
$$
have the same diagonal, $\operatorname{dim} \mathcal{N}_{\left(A-2 I_3\right)}$ is equal to three for the first, two for the second and one for the third. Calculate $\mathcal{N}_{\left(A-2 I_3\right)}$, for $j=1,2,3,4$ for each of the three choices of $A$.

Victor Salazar

Numerade Educator

01:52

Problem 10

Show that if $A \in \mathbb{F}^{n \times n}$ is a triangular matrix with entries $a_{i j}$, then $\sigma(A)=\cup_{i=1}^n\left\{a_{i i}\right\}$.
The cited theorems actually imply a little more:

Nick Johnson

Numerade Educator

02:37

Problem 11

Verify Theorem 4.4.

Nick Johnson

Numerade Educator

Problem 12

Verify Lemma 4.5 .

Check back soon!

03:34

Problem 13

Henry Carnick

Numerade Educator

Problem 14

Provide an example of three subspaces $\mathcal{U}, \mathcal{V}$ and $\mathcal{W}$ of a vector space $\mathcal{Y}$ over $\mathbb{F}$ such that $\mathcal{U}+\mathcal{V}=\mathcal{Y}$, but $\mathcal{W} \neq(\mathcal{W} \cap \mathcal{U}) \dot{+}(\mathcal{W} \cap \mathcal{V})$. [HINT: Simple examples exist with $\mathcal{Y}=\mathbb{R}^2$.]

If $\mathcal{U}_j, j=1, \ldots, k$, are finite dimensional subspaces of a vector space $\mathcal{Y}$ over $\mathbb{F}$, then the sum
(4.10) $\mathcal{U}_1+\cdots+\mathcal{U}_k=\left\{\mathbf{u}_1+\cdots+\mathbf{u}_k: \mathbf{u}_i \in \mathcal{U}_i\right.$ for $\left.i=1, \ldots, k\right\}$
is said to be direct if
$$
\operatorname{dim} \mathcal{U}_1+\cdots+\operatorname{dim} \mathcal{U}_k=\operatorname{dim}\left\{\mathcal{U}_1+\cdots+\mathcal{U}_k\right\} .
$$

If $\mathcal{U}=\mathcal{U}_1+\cdots+\mathcal{U}_k$ and the sum is direct, then we write
$$
\mathcal{U}=\mathcal{U}_1+\cdots+\mathcal{U}_k .
$$

If $k=2$, then formula (2.16) implies that the sum $\mathcal{U}_1+\mathcal{U}_2$ is direct if and only if $\mathcal{U}_1 \cap \mathcal{U}_2=\{0\}$. Therefore, the characterization (4.11) is consistent with the definition of the direct sum of two subspaces given earlier.

Check back soon!

05:51

Problem 15

Give an example of three subspaces $\mathcal{U}, \mathcal{V}$ and $\mathcal{W}$ of $\mathbb{R}^3$ such that $\mathcal{U} \cap \mathcal{V}=\{0\}, \mathcal{U} \cap \mathcal{W}=\{0\}$ and $\mathcal{V} \cap \mathcal{W}=\{0\}$ yet the $\operatorname{sum} \mathcal{U}+\mathcal{V}+\mathcal{W}$ is not direct.

Mengchun Cai

Numerade Educator

03:18

Problem 16

Let $\mathcal{Y}$ be a finite dimensional vector space over $\mathbb{F}$. Show that if $\mathcal{Y}=\mathcal{U} \dot{\mathcal{V}}$ and $\mathcal{V}=\mathcal{X} \dot{+} \mathcal{W}$, then $\mathcal{Y}=\mathcal{U} \dot{+} \mathcal{X} \dot{\mathcal{W}}$.

William Mead

Numerade Educator

02:38

Problem 17

Let $\mathcal{U}=\operatorname{span}\left\{\mathbf{u}_1, \ldots, \mathbf{u}_k\right\}$ over $\mathbb{F}$ and let $\mathcal{U}_j=\left\{\alpha \mathbf{u}_j: \alpha \in\right.$ $\mathbb{F}\}$. Show that the set of vectors $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_k\right\}$ is a basis for the vector space $\mathcal{U}$ over $F$ if and only if $\mathcal{U}_1 \dot{+} \cdots \dot{+} \mathcal{U}_k=\mathcal{U}$.

Cory Glover

Numerade Educator

04:38

Problem 18

Verify the corollary.
Formula (4.12) is extremely useful. In particular, it implies that
$$
\begin{aligned}
& A^2=\left(U D U^{-1}\right)\left(U D U^{-1}\right)=U D^2 U^{-1}, \\
& A^3=U D^3 U^{-1} \\
& \text { etc. }
\end{aligned}
$$

Victor Salazar

Numerade Educator

Problem 19

Show that if a matrix $A \in \mathbb{F}^{n \times n}$ is diagonalizable, i.e., if $A=U D U^{-1}$ with $D=\operatorname{diag}\left\{\lambda_1, \ldots, \lambda_n\right\}$, and if
$U=\left[\begin{array}{lll}\mathbf{u}_1 & \cdots & \mathbf{u}_n\end{array}\right]$ and $U^{-1}=\left[\begin{array}{c}\overrightarrow{\mathbf{v}_1} \\ \vdots \\ \overrightarrow{\mathbf{v}}_n\end{array}\right]$, then :
(1) $A^k=U D^k U^{-1}=\sum_{j=1}^n \lambda_j^k \mathbf{u}_j \overrightarrow{\mathbf{v}}_j$.
(2) $\left(A-\lambda I_n\right)^{-1}=U\left(D-\lambda I_n\right)^{-1} U^{-1}=\sum_{j=1}^n\left(\lambda_j-\lambda\right)^{-1} \mathbf{u}_j \overrightarrow{\mathbf{v}}_j$, if $\lambda \notin \sigma(A)$.

Check back soon!

View

Problem 20

Find an invertible matrix $U \in \mathbb{C}^{3 \times 3}$ and a diagonal matrix $D \in \mathbb{C}^{3 \times 3}$ so that $A=U D U^{-1}$ when $A$ is chosen equal to the matrix in the preceding example. [HINT: Follow the steps in the algorithm presented in the previous section.]

Victor Salazar

Numerade Educator

10:08

Problem 21

Find an invertible matrix $U$ such that $U^{-1} A U$ is equal to a diagonal matrix $D$ for each of the following two choices of $A$ :
$$
\left[\begin{array}{lll}
1 & 1 & 1 \\
0 & 2 & 2 \\
0 & 0 & 3
\end{array}\right],\left[\begin{array}{lll}
1 & 1 & 2 \\
0 & 2 & 2 \\
0 & 0 & 1
\end{array}\right] \text {. }
$$

Victor Salazar

Numerade Educator

04:00

Problem 22

Repeat Exercise 4.21 for
$$
A=\left[\begin{array}{lll}
1 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 1
\end{array}\right]
$$

Lucas Finney

Numerade Educator

Problem 23

Calculate $\operatorname{dim} \mathcal{N}_{\left(B_{\lambda_1}-\lambda_1 I_{13}\right),}$ for $j=1,2, \ldots$ when$$
B_{\lambda_1}=\left[\begin{array}{ccccc:cccc:cc:c:c}
\lambda_1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \lambda_1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \lambda_1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \lambda_1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \lambda_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\hdashline 0 & 0 & 0 & 0 & 0 & \lambda_1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \lambda_1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \lambda_1 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \lambda_1 & 0 & 0 & 0 & 0 \\
\hdashline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \lambda_1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \lambda_1 & 0 & 0 \\
\hdashline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \lambda_1 & 0 \\
\hdashline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \lambda_1
\end{array}\right]
$$
and build an array of symbols $\times$ with $\operatorname{dim} \mathcal{N}_{\left(B_{\lambda_1}-\lambda_1 I_{13}\right)^i}-\operatorname{dim} \mathcal{N}_{\left(B_{\lambda_1}-\lambda_1 I_{13}\right)^{i-1}}$ symbols $\times$ in the $i$ 'th row for $i=1,2, \ldots$. Check that the number of fundamental Jordan cells in $B_{\lambda_1}$ of size $i \times i$ is equal to the number of columns of height $i$ in the array corresponding to $\lambda_j$.

Check back soon!

Problem 24

Show that if
$$
A=U C_\alpha^{(n)} U^{-1}, \text { then } \operatorname{dim} \mathcal{N}_{\left(A-\lambda I_n\right)}=\left\{\begin{array}{lll}
0 & \text { if } & \lambda \neq \alpha \\
1 & \text { if } & \lambda=\alpha
\end{array} .\right.
$$

Check back soon!

Problem 25

Calculate $\operatorname{dim} \mathcal{N}_{\left(A-\lambda I_p\right)^t}$ for every $\lambda \in \mathbb{C}$ and $t=1,2, \ldots$ for the $26 \times 26$ matrix $A=U J U^{-1}$ when $J=\operatorname{diag}\left\{B_{\lambda_1}, B_{\lambda_2}, B_{\lambda_3}\right\}$, the points $\lambda_1, \lambda_2, \lambda_3$ are distinct, $B_{\lambda_1}$ is as in Exercise $4.23, B_{\lambda_2}=\operatorname{diag}\left\{C_{\lambda_2}^{(3)}, C_{\lambda_2}^{(3)}\right\}$ and $B_{\lambda_3}=\operatorname{diag}\left\{C_{\lambda_3}^{(4)}, C_{\lambda_3}^{(2)}, C_{\lambda_3}^{(1)}\right\}$. Build an array of symbols $\times$ for each eigenvalue $\lambda_j$ with $\operatorname{dim} \mathcal{N}_{\left(A-\lambda_j I_p\right)^i}-\operatorname{dim} \mathcal{N}_{\left(A-\lambda_j I_p\right)^{i-1}}$ symbols $\times$ in the $i$ 'th row for $i=1,2, \ldots$ and check that the number of fundamental Jordan cells in $B_{\lambda_j}$ of size $i \times i$ is equal to the number of columns in the array of height $i$.

Check back soon!

02:18

Problem 26

Find a pair of matrices $A$ and $B$ for which the formula (4.20) fails.

David Mccaslin

Numerade Educator

Problem 27

Complete the proof of the assertion in the preceding remark when $k=3$.

Check back soon!

01:25

Problem 28

If $B \in \mathbb{F}^{n \times n}$, then $\mathcal{R}_{B^n} \cap \mathcal{N}_{B^n}=\{0\}$. Show by example that the vector space
$$
\mathcal{R}_B \cap \mathcal{N}_B
$$
may contain nonzero vectors.

Carson Merrill

Numerade Educator

07:08

Problem 29

Show that if $A \in \mathbb{C}^{n \times n}$ has exactly two distinct eigenvalues in $\mathbb{C}$, then
$$
\mathcal{R}_{\left(A-\lambda_1 I_n\right)^n} \cap \mathcal{R}_{\left(A-\lambda_2 I_n\right)^n}=\{\mathbf{0}\} .
$$

Tamara Worner

Numerade Educator

Problem 30

Show that if $A \in \mathbb{C}^{n \times n}$ has exactly $k$ distinct eigenvalues $\lambda_1, \ldots, \lambda_k$ in $\mathbb{C}$ with algebraic multiplicities $\alpha_1, \ldots, \alpha_k$, then
$$
\mathcal{N}_{\left(A-\lambda_1 I_n\right)^{\alpha_1}}+\cdots+\mathcal{N}_{\left(A-\lambda_k I_n\right)^{\alpha_k}}=\mathbb{C}^n .
$$

Is it possible to reduce the powers further? Explain your answer.

Check back soon!

Problem 31

Verify formula (4.15). [HINT: In case of difficulty, start modestly by showing that if $B=\operatorname{diag}\left\{B_1, B_2, B_3\right\}$, then
$$
\left.\operatorname{dim} \mathcal{N}_B=\operatorname{dim} \ddot{\mathcal{N}}_{B_1}+\operatorname{dim} \mathcal{N}_{B_2}+\operatorname{dim} \mathcal{N}_{B_3} .\right]
$$

Check back soon!

Problem 32

Let $A$ be an $n \times n$ matrix.
(a): Show that if $\mathbf{u}_1, \ldots, \mathbf{u}_k$ are eigenvectors corresponding to distinct eigenvalues $\lambda_1, \ldots, \lambda_k$, then the vectors $\mathbf{u}_1, \ldots, \mathbf{u}_k$ are linearly independent. (Try to give a simple direct proof that exploits the fact that $\left.\left(A-\lambda_1 I_n\right) \cdots\left(A-\lambda_j I_n\right) \mathbf{u}_i=\left(\lambda_i-\lambda_1\right) \cdots\left(\lambda_i-\lambda_j\right) \mathbf{u}_i.\right)$
(b): Use the conclusions of part (a) to show that if $A$ has $n$ distinct eigenvalues, then $A$ is diagonalizable.

Check back soon!

03:02

Problem 33

Let $\mathbf{u} \in \mathbb{C}^n, \mathbf{v} \in \mathbb{C}^n$ and $B \in \mathbb{C}^{n \times n}$ be such that $B^4 \mathbf{u}=0$, $B^4 \mathbf{v}=0$ and the pair of vectors $B^3 \mathbf{u}$ and $B^3 \mathbf{v}$ are linearly independent in $\mathbb{C}^n$. Show that the eight vectors $\mathbf{u}, B \mathbf{u}, B^2 \mathbf{u}, B^3 \mathbf{u}, \mathbf{v}, B \mathbf{v}, B^2 \mathbf{v}$ and $B^3 \mathbf{v}$ are linearly independent in $\mathbb{C}^n$.

R M

Numerade Educator

03:02

Problem 34

Let $\mathbf{u} \in \mathbb{C}^n, \mathbf{v} \in \mathbb{C}^n$ and $B \in \mathbb{C}^{n \times n}$ be such that $B^4 \mathbf{u}=0$, $B^3 \mathbf{v}=0$ and the pair of vectors $B^3 \mathbf{u}$ and $B^2 \mathbf{v}$ are linearly independent in $\mathbb{C}^n$. Show that the seven vectors $\mathbf{u}, B \mathbf{u}, B^2 \mathbf{u}, B^3 \mathbf{u}, \mathbf{v}, B \mathbf{v}$ and $B^2 \mathbf{v}$ are linearly independent in $\mathbb{C}^n$.

R M

Numerade Educator

03:51

Problem 35

Let $B \in \mathbb{C}^{n \times n}$. Show that $\mathcal{N}_B \subseteq \mathcal{N}_{B^2} \subseteq \mathcal{N}_{B^3} \subseteq \cdots$ and that if $\mathcal{N}_{B^j}=\mathcal{N}_{B^{j+1}}$ for $j=k$, then the equality prevails for every integer $j>k$ also.

Srilakshmi E K

Numerade Educator

02:39

Problem 36

Show that if $B \in \mathbb{C}^{n \times n}$, then $\operatorname{dim} \mathcal{N}_{B^2} \leq 2 \operatorname{dim} \mathcal{N}_B$. [REMARK: The correct way to interpret this is: $\operatorname{dim} \mathcal{N}_{B^2}-\operatorname{dim} \mathcal{N}_B \leq \operatorname{dim} \mathcal{N}_B$.]

Victor Salazar

Numerade Educator

02:15

Problem 37

Calculate $\left[\begin{array}{ll}a & 1 \\ 0 & a\end{array}\right]^{100}$. [HINT: To see the pattern, write the given matrix as $a I_2+F$ and note that since $F^2=0,\left(a I_2+F\right)^2,\left(a I_2+\right.$ $F)^3, \ldots$, have a simple form.]

Nick Johnson

Numerade Educator