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Linear Algebra in Action

Harry Dym

Chapter 6

Calculating Jordan forms - all with Video Answers

Educators

Chapter Questions

Problem 1

Verify assertion (3) in Lemma 6.3.

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05:44

Problem 2

Let $A \in \mathbb{C}^{n \times n}$ be similar to a Jordan matrix $J$ that contains exactly $k_j$ Jordan cells $C_\mu^{(j)}$ of size $j \times j$ with $\mu$ on the diagonal for $j=1, \ldots, \ell$ and let $B=A-\mu I_n$. Show that formula (6.3) for $k_j$ is still valid.

Tim Strang
Tim Strang
Numerade Educator

Problem 3

Calculate $\operatorname{dim} \mathcal{N}_B$, for $j=1, \ldots, 15$, when $B=B_{\lambda_1}-\lambda_1 I_{15}$ and
$$
B_{\lambda_1}=\operatorname{diag}\left\{C_{\lambda_1}^{(5)}, C_{\lambda_1}^{(3)}, C_{\lambda_1}^{(3)}, C_{\lambda_1}^{(2)}, C_{\lambda_1}^{(1)}, C_{\lambda_1}^{(1)}\right\}
$$

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03:03

Problem 4

Find an $11 \times 11$ matrix $B$ such that $\operatorname{dim} \mathcal{N}_B=4, \operatorname{dim} \mathcal{N}_{B^2}=$ $7, \operatorname{dim} \mathcal{N}_{B^3}=9, \operatorname{dim} \mathcal{N}_{B^4}=10$ and $\operatorname{dim} \mathcal{N}_{B^5}=11$.

Kushal Mohnot
Kushal Mohnot
Numerade Educator

Problem 5

Let $A \in \mathbb{C}^{n \times n}$ be similar to a Jordan matrix $J$ that contains exactly $k_1$ Jordan cells $C_\mu^{(1)}, k_2$ Jordan cells $C_\mu^{(2)}, \ldots, k_{\ell}$ Jordan cells $C_\mu^{(\ell)}$ with $\mu$ on the diagonal and let $B=A-\mu I_n$. Show that
(6.4) $\operatorname{dim} \mathcal{N}_{B^j}=\left\{\begin{array}{ccc}k_1+k_2+\cdots+k_{\ell} & \text { if } & j=1 \\ k_1+2 k_2+\cdots+(j-1) k_{j-1}+j \sum_{i=j}^{\ell} k_i & \text { if } & j \geq 2 .\end{array}\right.$

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Problem 6

Show that in the setting of Exercise 6.5
(6.5) $\operatorname{dim} \mathcal{N}_{B^{j+1}}-\operatorname{dim} \mathcal{N}_{B^j}=k_{j+1}+\cdots+k_{\ell} \quad$ for $\quad j=1, \ldots, n-1$.

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Problem 7

Verify that the $\ell_4+\ell_3+\ell_2+\ell_1$ vectors exhibited in the array just above are linearly independent.

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07:23

Problem 8

Show that in the array exhibited just above the set of vectors in the first $k$ rows is a basis for $\mathcal{N}_{B^k}$ for $k=1,2,3,4$; i.e., the set of vectors in the first row is a basis for $\mathcal{N}_B$, the set of vectors in the first two rows is a basis for $\mathcal{N}_{B^2}$, etc.

Anthony Ramos
Anthony Ramos
Numerade Educator
10:02

Problem 9

Find a Jordan form $J$ and an invertible matrix $U$ such that
$$
\left[\begin{array}{lllll}
2 & 0 & 0 & 0 & 2 \\
0 & 2 & 0 & 0 & 0 \\
0 & 2 & 2 & 0 & 0 \\
0 & 0 & 0 & 2 & 0 \\
0 & 0 & 0 & 0 & 2
\end{array}\right]=U J U^{-1} .
$$

Tim Strang
Tim Strang
Numerade Educator
14:02

Problem 10

Find a Jordan form $J$ and an invertible matrix $U$ such that
$$
\left[\begin{array}{llll}
1 & 2 & 0 & 0 \\
0 & 1 & 2 & 0 \\
0 & 0 & 1 & 2 \\
0 & 0 & 0 & 1
\end{array}\right]=U J U^{-1} .
$$

Tim Strang
Tim Strang
Numerade Educator
14:02

Problem 11

Find a Jordan form $J$ and an invertible matrix $U$ such that
$$
\left[\begin{array}{llll}
1 & 2 & 0 & 0 \\
0 & 1 & 0 & 0 \\
x & 0 & 1 & 0 \\
0 & 0 & 2 & 1
\end{array}\right]=U J U^{-1},
$$
first for $x=0$ and then for $x=1$.

Tim Strang
Tim Strang
Numerade Educator
14:02

Problem 12

Find a Jordan form $J$ and an invertible matrix $U$ such that
$$
A=\left[\begin{array}{lllllll}
2 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 3 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 3 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 3 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 3 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 3 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 2
\end{array}\right]=U J U^{-1}
$$

Tim Strang
Tim Strang
Numerade Educator
14:02

Problem 13

Find a Jordan form $J$ and an invertible matrix $U$ such that
$$
A=\left[\begin{array}{ccccc}
1 & 1 & 0 & 0 & 0 \\
1 & 2 & 1 & 0 & 0 \\
13 & -8 & 6 & 0 & 0 \\
-11 & 8 & -5 & 1 & 1 \\
-22 & 17 & -12 & -4 & 5
\end{array}\right]=U J U^{-1}
$$

Tim Strang
Tim Strang
Numerade Educator
04:15

Problem 14

Show that the two conditions (6.7) and (6.8) are equivalent.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:08

Problem 15

Let $A \in \mathbb{R}^{n \times n}$ and suppose that $n \geq 2$. Show that:
(1) There exists a one-dimensional subspace $\mathcal{U}$ of $\mathbb{C}^n$ that is invariant under $A$.
(2) There exists a subspace $\mathcal{V}$ of $\mathbb{R}^n$ of dimension less than or equal to two that is invariant under $A$.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 16

Verify formula (6.12) when $m_j=4$.
A matrix of the form $V=\left[\begin{array}{lll}V_1 & \cdots & V_k\end{array}\right]$, with $V_j$ as in (6.11) is called a generalized Vandermonde matrix.

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Problem 17

Verify Corollary 6.11.

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04:41

Problem 18

Verify that the matrix identity in Example 6.12 is correct.

James Kiss
James Kiss
Numerade Educator

Problem 19

Let $A \in \mathbb{C}^{n \times n}$ have $k$ distinct eigenvalues $\lambda_1, \ldots, \lambda_k$ with geometric multiplicities $\gamma_1, \ldots, \gamma_k$ and algebraic multiplicities $\alpha_1, \ldots, \alpha_k$, respectively. Show that if $\gamma_j=1$ for $j=1, \ldots, k$, then $A$ is similar to a companion matrix $S_f$ based on a polynomial $f(\lambda)$ and find $f(\lambda)$.

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00:50

Problem 20

Show that if $A \in \mathbb{C}^{n \times n}$ is similar to the Jordan matrix
$$
J=\operatorname{diag}\left\{C_{\lambda_1}^{(4)}, C_{\lambda_1}^{(2)}, C_{\lambda_2}^{(3)}, C_{\lambda_2}^{(1)}, C_{\lambda_3}^{(3)}\right\},
$$
then $A$ is also similar to the block diagonal matrix diag $\left\{S_{g_1}, S_{g_2}\right\}$ based on a pair of polynomials $g_1(\lambda)$ and $g_2(\lambda)$ and find the polynomials.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 21

Find a Jordan form for $\mu\left(I_n-\mu C_0^{(n)}\right)^{-1}$ when $\mu \neq 0$.

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10:02

Problem 22

Find a Jordan form $J$ for the matrix $A=\left[\begin{array}{lllll}2 & 2 & 1 & 0 & 0 \\ 0 & 2 & 3 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 3 & 1 & 2 & 0 \\ 0 & 1 & 1 & 1 & 2\end{array}\right]$.

Tim Strang
Tim Strang
Numerade Educator
14:07

Problem 23

Find an invertible matrix $U \in \mathbb{C}^{5 \times 5}$ such that $A=U J U^{-1}$ for the matrix $A$ with Jordan form $J$ that was considered in Exercise 6.22.

Tim Strang
Tim Strang
Numerade Educator

Problem 24

Let $B \in \mathbb{C}^{n \times n} ;$ let $\mathbf{u}_1 \in \mathcal{N}_B ; \mathbf{v}_1, \mathbf{v}_2 \in \mathcal{N}_{B^2} ; \mathbf{w}_1, \mathbf{w}_2 \in \mathcal{N}_{B^3}$; and assume that the 5 vectors $B^2 \mathbf{w}_1, B^2 \mathbf{w}_2, B \mathbf{v}_1, B \mathbf{v}_2, \mathbf{u}_1$ are linearly independent over $\mathbb{C}$. Show that the 11 vectors $B^2 \mathbf{w}_1, B \mathbf{w}_1, \mathbf{w}_1, B^2 \mathbf{w}_2$, $B \mathbf{w}_2, \mathbf{w}_2, B \mathbf{v}_1, \mathbf{v}_1, B \mathbf{v}_2, \mathbf{v}_2, \mathbf{u}_1$ are also linearly independent over $\mathbb{C}$.

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14:02

Problem 25

Find an invertible matrix $U$ such that $U^{-1} A U$ is in Jordan form when $A=\left[\begin{array}{ccc}1 & 0 & i \\ 0 & 2 & 0 \\ -i & 0 & 1\end{array}\right]$. [NOTE: $i=\sqrt{-1}$.]

Tim Strang
Tim Strang
Numerade Educator
05:11

Problem 26

Find a Jordan form $J$ for the matrix
$$
A=\left[\begin{array}{ccccc}
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
8 & -12 & 6 & 0 & 0 \\
-1 & 1 & 0 & 0 & 1 \\
-4 & 1 & 0 & -4 & 4
\end{array}\right] .
$$

Tim Strang
Tim Strang
Numerade Educator
View

Problem 27

Find an invertible matrix $U$ such that $A U=U J$ for the matrices $A$ and $J$ considered in Exercise 6.26.
The next three exercises are adapted from [58].

Victor Salazar
Victor Salazar
Numerade Educator
02:34

Problem 28

Show that if $n \geq 2$, then the matrix $\left(C_\mu^{(n)}\right)^2$ is similar to the matrix $C_{\mu^2}^{(n)}$ if and only if $\mu \neq 0$.

Nick Johnson
Nick Johnson
Numerade Educator
01:20

Problem 29

Let $B \in \mathbb{C}^{p \times p}$ be a triangular matrix with diagonal entries $b_{i i}=\lambda \neq 0$ for $i=1, \ldots, p$ and let $V \in \mathbb{C}^{p \times p}$. Show that $B^2 V=V B^2 \Longleftrightarrow$ $B V=V B$.

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 30

Let $A, B \in \mathbb{C}^{n \times n}$ and suppose that the eigenvalues of $A$ and $B$ are nonnegative and that $\mathcal{N}_A=\mathcal{N}_{A^2}$ and $\mathcal{N}_B=\mathcal{N}_{B^2}$. Show that $A^2=B^2 \Longleftrightarrow A=B$.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 31

Let $A \in \mathbb{C}^{n \times n}$ be a companion matrix and let $p(\lambda)=$ $\operatorname{det}\left(\lambda I_n-A\right)$. Show that
(6.13) $A\left[\begin{array}{c}1 \\ \lambda \\ \vdots \\ \lambda^{n-1}\end{array}\right]=\left[\begin{array}{c}\lambda \\ \lambda^2 \\ \vdots \\ \lambda^n-p(\lambda)\end{array}\right], A\left[\begin{array}{c}0 \\ 1 \\ \vdots \\ (n-1) \lambda^{n-2}\end{array}\right]=\left[\begin{array}{c}1 \\ 2 \lambda \\ \vdots \\ n \lambda^{n-1}-p^{\prime}(\lambda)\end{array}\right]$
and differentiate once again with respect to $\lambda$ to obtain the next term in the indicated sequence of formulas.

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05:10

Problem 32

Find an invertible matrix $U$ and a matrix $J$ in Jordan form such that $A=U J U^{-1}$ if $A \in \mathbb{C}^{6 \times 6}$ is a companion matrix, $\operatorname{det}\left(\lambda I_6-A\right)=$ $\left(\lambda-\lambda_1\right)^4\left(\lambda-\lambda_2\right)^2$ and $\lambda_1 \neq \lambda_2$.

Tim Strang
Tim Strang
Numerade Educator

Problem 33

Let $A \in \mathbb{C}^{n \times n}$ with $k$ distinct eigenvalues $\lambda_1, \ldots, \lambda_k$. Show that if the geometric multiplicity $\gamma_j$ of $\lambda_j$ is equal to one for $j=1, \ldots, k$, then $A$ is similar to a companion matrix.

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