Chapter 19, Realization theory Video Solutions, Linear Algebra in Action

Problem 1

Check that the mvf $F(\lambda)$ defined by formula (19.3) does not change if $C$ is replaced by $C S, A$ by $S^{-1} A S$ and $B$ by $S^{-1} B$ for some invertible matrix $S \in \mathbb{C}^{n \times n}$.

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Problem 2

Show that if $F_1(\lambda)=D_1+C_1\left(\lambda I_{n_1}-A_1\right)^{-1} B_1$ is a $p \times q$ mvf and $F_2(\lambda)=D_2+C_2\left(\lambda I_{n_2}-A_2\right)^{-1} B_2$ is a $q \times r$ mvf, then
$$
F_1(\lambda) F_2(\lambda)=D_3+C_3\left(\lambda I_n-A_3\right)^{-1} B_3 \text {, }
$$
where
$$
D_3=D_1 D_2, \quad C_3=\left[\begin{array}{ll}
C_1 & D_1 C_2
\end{array}\right], \quad A_3=\left[\begin{array}{cc}
A_1 & -B_1 C_2 \\
O & A_2
\end{array}\right], \quad B_3=\left[\begin{array}{c}
B_1 D_2 \\
B_2
\end{array}\right]
$$
and $n=n_1+n_2$.

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Problem 3

Show that if $D \in \mathbb{C}^{p \times p}$ is invertible, then (19.5)
$$
\left\{D+C\left(\lambda I_n-A\right)^{-1} B\right\}^{-1}=D^{-1}-D^{-1} C\left(\lambda I_n-\left[A-B D^{-1} C\right]\right)^{-1} B D^{-1} \text {. }
$$

Let $C \in \mathbb{C}^{p \times n}, A \in \mathbb{C}^{n \times n}$ and $B \in \mathbb{C}^{n \times q}$. Then the pair $(A, B)$ is said to be controllable if the controllability matrix
$$
\mathfrak{C}=\left[\begin{array}{llll}
B & A B & \cdots & A^{n-1} B
\end{array}\right]
$$
is right invertible, i.e., if
$$
\operatorname{rank} \mathfrak{C}=n
$$
The pair $(C, A)$ is said to be observable if the observability matrix
$$
\mathfrak{O}=\left[\begin{array}{c}
C \\
C A \\
\vdots \\
C A^{n-1}
\end{array}\right]
$$
is left invertible, i.e., if its null space
$$
\mathcal{N}_{\mathfrak{D}}=\{\mathbf{0}\}
$$

Nick Johnson

Numerade Educator

Problem 4

Show that $(C, A)$ is an observable pair if and only if the pair $\left(A^H \cdot C^H\right)$ is controllable.

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Problem 5

Show that $(A, B)$ is controllable if and only if condition (4) in Lemma 19.2 is met.

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Problem 6

Show that $(A, B)$ is controllable if and only if condition (5) in Lemma 19.2 is met.

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04:08

Problem 7

Show that $(A, B)$ is controllable if and only if $\mathfrak{C C}^H$ is invertible.

Michael Jacobsen

Numerade Educator

Problem 8

Verify the equivalence (19.6) and then complete the proof that (3) is equivalent to (1) in Lemma 19.3.

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Problem 9

Show that in Lemma 19.3, (4) is equivalent to (1).

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Problem 10

Show that in Lemma 19.3, (5) is equivalent to (1).

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06:28

Problem 11

Show that the pair $(C, A)$ is observable if and only if the matrix $\mathcal{O}^H \mathcal{O}$ is invertible.

Jimmy Yao

Numerade Educator

Problem 12

Let $F(\lambda)=I_p+C\left(\lambda I_n-A\right)^{-1} B$ and $G(\lambda)=I_p-C_1\left(\lambda I_n-\right.$ $\left.A_1\right)^{-1} B_1$. Show that if $C_1=C$ and $(C, A)$ is an observable pair, then $F(\lambda) G(\lambda)=I_p$ if and only if $B_1=B$ and $A_1=A-B C$.

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Problem 13

Verify the asserted uniqueness of the invertible matrix $Y$ that is constructed in the proof of (2) of Theorem 19.4.

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Problem 14

Verify the assertions in formula (19.8).

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Problem 15

Complete the proof of Lemma 19.6.

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Problem 16

Let $(C, A) \in \mathbb{C}^{p \times n} \times \mathbb{C}^{n \times n}$ be an observable pair and let $\mathbf{u} \in \mathbb{C}^n$. Show that $C\left(\lambda I_n-A\right)^{-1} \mathbf{u}$ has a pole at $\alpha$ if and only if $\left(\lambda I_n-A\right)^{-1} \mathbf{u}$ has a pole at $\alpha$. [HINT: First show that it suffices to focus on the case that $A=C_\alpha^{(n)}$ is a single Jordan cell.]

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Problem 17

Let $(C, A) \in \mathbb{C}^{p \times n} \times \mathbb{C}^{n \times n}$ be an observable pair and let $\mathbf{u}(\lambda)=\mathbf{u}_0+\lambda \mathbf{u}_1+\cdots+\lambda^k \mathbf{u}_k$ be a vector polynomial with coefficients in $\mathbb{C}^n$. Show that $C\left(\lambda I_n-A\right)^{-1} \mathbf{u}(\lambda)$ has a pole at $\alpha$ if and only if $\left(\lambda I_n-A\right)^{-1} \mathbf{u}(\lambda)$ has a pole at $\alpha$. [HINT: Try Exercise 19.16 first to warm up.]

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Problem 18

Show that the coefficients $F_{\ell}^{(j)}$ of $\left(\lambda-\lambda_j\right)^{-\ell}$ in the Laurent expansion of the matrix valued function $F(\lambda)$ considered in Theorem 19.9 are given by the formula
$$
F_{\ell}^{(j)}=C P_j\left(A-\lambda_j I_n\right)^{\ell-1} B \text { for } \ell=1, \cdots, \alpha_j,
$$
where $P_j=U_j V_j$ is the projector defined in the proof of the theorem.

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Problem 19

Show that the projector $P_j$ defined in the proof of Theorem 19.9 is the Riesz projector that is defined by the formula
$$
P_j=\frac{1}{2 \pi i} \int_{\Gamma_j}\left(\lambda I_n-A\right)^{-1} d \lambda,
$$
if $\Gamma_j$ is a small enough circle centered at $\lambda_j$ and directed counterclockwise.

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Problem 20

Show that if $C \in \mathbb{C}^{p \times n}$ and $A \in \mathbb{C}^{n \times n}$, then
$$
\operatorname{rank}\left[\begin{array}{l}
C \\
C A \\
\vdots \\
C A^{\ell}
\end{array}\right]=\operatorname{rank}\left[\begin{array}{l}
C \\
C\left(\alpha I_n+A\right) \\
C\left(\alpha I_n+A\right)^{\ell}
\end{array}\right]
$$
for every positive integer $\ell$ and every point $\alpha \in \mathbb{C}$.

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Problem 21

Show that if $A \in \mathbb{C}^{n \times n}$ and $B \in \mathbb{C}^{n \times q}$, then
$$
\operatorname{rank}\left[B A B \cdots A^{\ell} B\right]=\operatorname{rank}\left[B\left(\alpha I_n+A\right) B \cdots\left(\alpha I_n+A\right)^{\ell} B\right]
$$
for every positive integer $\ell$ and every point $\alpha \in \mathbb{C}$.

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Problem 22

Show that $(A, B)$ is stabilizable if and only if $\left(B^H, A^H\right)$ is detectable.

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Problem 23

Show that if $(C, A)$ is detectable and $(A, B)$ is stabilizable, then there exist matrices $K$ and $L$ such that
$$
\sigma\left(\left[\begin{array}{cc}
A & -B K \\
L C & A-B K-L C
\end{array}\right]\right) \subset \Pi_{-} .
$$

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07:29

Problem 24

Justify assertion (3) of Lemma 19.12.

Ernest Castorena

Numerade Educator

02:50

Problem 25

Verify formulas (19.13) and (19.14) and check that the inner product in the latter is well defined; i.e., if $F X \mathbf{u}_1=F X \mathbf{u}_2$ and $F X \mathbf{v}_1=F X \mathbf{v}_2$, then $\mathbf{v}_1^H X \mathbf{u}_1=\mathbf{v}_2^H X \mathbf{u}_2$.

James Kiss

Numerade Educator

01:13

Problem 26

Show that if $J \in \mathbb{C}^{m \times m}$ is a signature matrix, then either $J= \pm I_m$ or $J=U$ diag $\left\{I_p,-I_q\right\} U^H$, with $U$ unitary, $p \geq 1, q \geq 1$ and $p+q=m$.

Srilakshmi E K

Numerade Educator

04:59

Problem 27

Complete the proof of Lemma 19.16 by justifying the implications $(4) \Longrightarrow(1) \Longrightarrow(2)$.

Stanley Enemuo

Numerade Educator

02:10

Problem 28

Verify the statements in (19.32).

Manik Pulyani

Numerade Educator

Problem 29

Verify formula (19.34). [HINT: The trick in this calculation (and others of this kind) is to note that in the product, the two terms
$$
C\left(\lambda I_n-A\right)^{-1} X C^H J+C\left(\lambda I_n-A_1\right)^{-1} X_1 C^H J C\left(\lambda I_n-A\right)^{-1} X C^H J
$$
can be reexpressed as
$$
C\left(\lambda I_n-A_1\right)^{-1}\left\{\lambda I_n-A_1+X_1 C^H J C\right\}\left(\lambda I_n-A\right)^{-1} X C^H J,
$$
which simplifies beautifully.]

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Problem 30

Show that if $f \in \mathcal{M}_2$, then
$$
\left\|\vartheta_1 f\right\|_{\mathcal{M}}=\|f\|_{\mathcal{M}_2} .
$$

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Problem 31

Show that $\operatorname{rank} X_2=\operatorname{rank} X-\operatorname{rank} X_1$.

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01:12

Problem 32

Let $A \in \mathbb{C}^{n \times n}, C \in \mathbb{C}^{m \times n}, J \in \mathbb{C}^{m \times m}$, let $\lambda_1, \ldots, \lambda_k$ denote the distinct eigenvalues of $A$; and let $P$ be a solution of the Stein equation $P-A^H P A=C^H J C$. Show that if $1-\lambda_i \overline{\lambda_j} \neq 0$ for $i, j=1, \ldots, k$, then $\mathcal{N}_{\mathfrak{D}} \subseteq \mathcal{N}_P$. [HINT: $\mathcal{N}_{\mathcal{D}}$ is invariant under A.]

Manik Pulyani

Numerade Educator

Problem 33

Let $A \in \mathbb{C}^{n \times n}, C \in \mathbb{C}^{m \times n}, J \in \mathbb{C}^{m \times m}$; and let $P$ be a solution of the Lyapunov equation $A^H P+P A=C^H J C$. Show that if $\sigma(A) \cap \sigma\left(-A^H\right)=\emptyset$, then $\mathcal{N}_{\mathfrak{D}} \subseteq \mathcal{N}_P$.

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