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Linear Algebra in Action

Harry Dym

Chapter 18

Matrix equations - all with Video Answers

Educators

Chapter Questions

Problem 1

Verify the corollary.

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Problem 2

Let $A=C_\alpha^{(2)}$ and $B=C_\beta^{(2)}$ and suppose that $\alpha \beta=1$. Show that the equation $X-A X B=C$ has no solutions if either $c_{21} \neq 0$ or $\alpha c_{11} \neq \beta c_{22}$.

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02:54

Problem 3

Let $A=C_\alpha^{(2)}$ and $B=C_\beta^{(2)}$ and suppose that $\alpha \beta=1$. Show that if $c_{21}=0$ and $\alpha c_{11}=\beta c_{22}$, then the equation $X-A X B=C$ has infinitely many solutions.

Gaurav Kalra
Gaurav Kalra
Numerade Educator
05:04

Problem 4

Find the unique solution $X \in \mathbb{C}^{p \times p}$ of equation (18.6) when $A=C_0^{(p)}, C=\mathbf{e}_1 \mathbf{u}^H+\mathbf{u e}_1^H+\mathbf{e}_p \mathbf{e}_p^H$ and $\mathbf{u}^H=\left[\begin{array}{llll}0 & \overline{s_1} & \ldots & \overline{s_{p-1}}\end{array}\right]$.

Chris Trentman
Chris Trentman
Numerade Educator

Problem 5

Complete the proof of Theorem 18.5.

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04:56

Problem 6

Let $A \in \mathbb{C}^{n \times n}$. Show that the Lyapunov equation
$$
A^H X+X A=Q
$$
has a unique solution for each choice of $Q \in \mathbb{C}^{n \times n}$ if and only if $\sigma(A) \cap$ $\sigma\left(-A^H\right)=\emptyset$.

Ryan Williams
Ryan Williams
Numerade Educator

Problem 7

Let $A \in \mathbb{C}^{n \times n}$. Show that if $\sigma(A) \subset \Pi_{+}$, the open right half plane, then the equation $A^H X+X A=Q$ has a unique solution for every choice of $Q \in \mathbb{C}^{n \times n}$ and that this solution can be expressed as
$$
X=\int_0^{\infty} e^{-t A^H} Q e^{-t A} d t
$$
for every choice of $Q \in \mathbb{C}^{n \times n}$. [HINT: Integrate the formula
$$
A^H \int_0^{\infty} e^{-t A^H} Q e^{-t A} d t=-\int_0^{\infty} \frac{d}{d t}\left(e^{-t A^H} Q\right) e^{-t A} d t
$$
by parts.]

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Problem 8

Show that in the setting of Exercise 18.7, the solution $X$ can also be expressed as
$$
X=-\frac{1}{2 \pi} \int_{-\infty}^{\infty}\left(i \mu I_n+A^H\right)^{-1} Q\left(i \mu I_n-A\right)^{-1} d \mu .
$$

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Problem 9

Let $A=\operatorname{diag}\left\{A_{11}, A_{22}\right\}$ be a block diagonal matrix in $\mathbb{C}^{n \times n}$ with $\sigma\left(A_{11}\right) \subset \Pi_{+}$and $\sigma\left(A_{22}\right) \subset \Pi_{-}$, let $Q \in \mathbb{C}^{n \times n}$ and let $Y \in \mathbb{C}^{n \times n}$ and $Z \in \mathbb{C}^{n \times n}$ be solutions of the Lyapunov equation $A^H X+X A=Q$. Show that if $Y$ and $Z$ are written in block form consistent with the block decomposition of $A$, then $Y_{11}=Z_{11}$ and $Y_{22}=Z_{22}$.

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Problem 10

Let $A, Q \in \mathbb{C}^{n \times n}$. Show that if $\sigma(A) \cap i \mathbb{R}=\emptyset$ and if $Y$ and $Z$ are both solutions of the same Lyapunov equation $A^H X+X A=Q$ such that $Y-Z \succeq O$, then $Y=Z$.

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Problem 11

Let $A=\sum_{j=1}^3 \mathbf{e}_j \mathbf{e}_{j+1}^T=C_0^{(4)}$ and let $T$ denote the linear transformation from $\mathbb{C}^{4 \times 4}$ into itself that is defined by the formula $T X=$ $A^H X-X A$.
(a) Calculate $\operatorname{dim} \mathcal{N}_T$.
(b) Show that a matrix $X \in \mathbb{C}^{4 \times 4}$ with entries $x_{i j}$ is a solution of the matrix equation $A^H X-X A=\left[\begin{array}{cccc}0 & -a & -b & -c \\ a & 0 & 0 & 0 \\ b & 0 & 0 & 0 \\ c & 0 & 0 & 0\end{array}\right]$ if and only if $X$ is a Hankel matrix with $x_{11}=a, x_{12}=b$ and $x_{13}=c$.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 12

Let $A=C_0^{(4)}$ and let $T$ denote the linear transformation from $\mathbb{C}^{4 \times 4}$ into itself that is defined by the formula $T X=X-A^H X A$.
(a) Calculate $\operatorname{dim} \mathcal{N}_T$.
(b) Show that a matrix $X \in \mathbb{C}^{4 \times 4}$ is a solution of the matrix equation $X-A^H X A=\left[\begin{array}{llll}a & b & c & d \\ e & 0 & 0 & 0 \\ f & 0 & 0 & 0 \\ g & 0 & 0 & 0\end{array}\right]$ if and only if $X$ is a Toeplitz matrix.

Victor Salazar
Victor Salazar
Numerade Educator
04:59

Problem 13

Complete the proof of Theorem 18.7 by verifying the cases $p=0$ and $p=n$.

Stanley Enemuo
Stanley Enemuo
Numerade Educator

Problem 14

Verify the identity $S G S^{-1}=-G^H$ and the assertion of Lemma 18.8 .

If $\sigma(G) \cap i \mathbb{R}=\emptyset$, then Lemma 18.8 guarantees that $G$ admits a Jordan decomposition of the form
$$
G=U\left[\begin{array}{ll}
J_1 & O \\
O & J_2
\end{array}\right] U^{-1},
$$
where $J_1, J_2 \in \mathbb{C}^{n \times n}, \sigma\left(J_1\right) \subset \Pi_{-}$and $\sigma\left(J_2\right) \subset \Pi_{+}$.

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Problem 15

Let $A \in \mathbb{C}^{n \times n}, B \in \mathbb{C}^{n \times k}$. Show that if $\sigma(A) \cap \mathbb{R}=\emptyset$ and rank $\left[\begin{array}{ll}A-\lambda I_n & B\end{array}\right]=n$ for every point $\lambda \in \bar{\Pi}_{+}$, then there exists exactly one Hermitian solution $X$ of the Riccati equation $A^H X+X A-X B B^H X=O$ such that $\sigma\left(A-B B^H X\right) \subset \Pi_{\text {_ }}$.

For future applications, it will be convenient to have another variant of Theorem 18.13 .

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Problem 16

Show that if $N, Y, M \in \mathbb{C}^{n \times n}$, then the $n \times n$ matrix valued function
$$
X(t)=e^{t N} Y e^{t M}
$$
is a solution of the differential equation
$$
X^{\prime}(t)=N X(t)+X(t) M
$$
that meets the initial condition $X(0)=Y$.

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