MATH319 Exercises 3 Assessed Exercise 1 5 Assessed Exercise 2

4 Workshop Exercise 2

W2.1 Let $A$ be a complex $(n\times n)$ matrix with distinct eigenvalues $\lambda_{1},\dots,\lambda_{n}$ . Show that there exists an invertible matrix $S$ such that

(sI-A)^{-1}=S\left[\begin{pmatrix}{{1}\over{s-\lambda_{1}}}&0&\dots&0\cr 0&{{1% }\over{s-\lambda_{2}}}&\ddots&\vdots\cr\vdots&\ddots&\ddots&0\cr 0&\dots&0&{{1% }\over{s-\lambda_{n}}}\cr\end{pmatrix}\right]S^{-1}

for all $s\neq\lambda_{1},\dots,\lambda_{n}$ .

W2.2 (i) Find a SISO system that has transfer function

T(s)={{2s^{2}-3s+4}\over{s^{3}+5s^{2}+6s+7}}.

(ii) Find approximate numerical values for the eigenvalues of $A$ .

W2.3 Let $A$ be the matrix

A=\left[\begin{pmatrix}0&0&1&0\cr 0&0&0&1\cr a&b&0&0\cr c&d&0&0\cr\end{pmatrix% }\right].

(i) Find $\det A$ .

(ii) Use reduction of determinants to find $\det(sI-A)$ .

W2.4 Let $A$ be a $(n\times n)$ complex matrix, and for $B$ a $(n\times 1)$ column vector, let

V=\Bigl{\{}\sum_{j=0}^{n-1}a_{j}A^{j}B:a_{j}\in{\bf C};j=0,\dots,n-1\Bigr{\}}.

(i) Show that $V$ is a complex vector space.

(ii) Show that $V$ has dimension one, if and only if $B$ is an eigenvector of $A$ .

(iii) Use the Cayley–Hamilton theorem from frame 30 to show that, for all $A$ , the left multiplication $L_{A}:X\mapsto AX$ for $X\in V$ defines a linear operator $L_{A}:V\rightarrow V$ .

(iv) Given that dimension of $V$ equals the rank of $Q$ where

Q=\left[\begin{pmatrix}B&AB&\dots&A^{n-1}B\cr\end{pmatrix}\right],

calculate the dimension of $V$ when

A=\left[\begin{pmatrix}1&3&5&0\cr 0&1&9&6\cr 1&1&4&-7\cr 2&2&1&8\cr\end{% pmatrix}\right],\quad B=\left[\begin{pmatrix}1\cr-5\cr-1/2\cr 3\cr\end{pmatrix% }\right].

W2.5 Find a SISO system $(A,B,C,D)$ that has transfer function

T(s)={{2s^{3}+s^{2}-5s+1}\over{s^{4}-6s^{3}+5s^{2}+4s+2}}.

W2.6 Express the matrix

A=\left[\begin{pmatrix}1&1&-1\cr 4&1&1\cr 5&-2&4\end{pmatrix}\right]

in Jordan canonical form.

W2.7 Let $A$ be $(n\times n)$ , $B$ be $(n\times 1)$ , $C$ be $(1\times n)$ and $D$ be $(1\times 1)$ constant matrices. Show that

D+C(sI-A)^{-1}B={{\det\left[\begin{pmatrix}sI-A&B\cr-C&D\cr\end{pmatrix}\right% ]}\over{\det(sI-A)}}.

W2.8 Let $(A,B,C,D)$ be a $S I S O$ with transfer function $T$ . Show that $(A^{t},C^{t},B^{t},D)$ is also a $S I S O$ with transfer function $T$ , where here $A^{t}$ denotes the transpose of $A$ .

W2.9 Suppose that $S$ is an invertible matrix. From the $S I S O$ system $(A,B,C,D)$ , we introduce another $S I S O$ by $(\hat{A},\hat{B},\hat{C},\hat{D})=(S^{-1}AS,S^{-1}B,CS,D)$ .

(i) Show that

\det\left[\begin{pmatrix}sI-A&B\cr-C&D\cr\end{pmatrix}\right]=\det\left[\begin% {pmatrix}sI-\hat{A}&\hat{B}\cr-\hat{C}&\hat{D}\cr\end{pmatrix}\right].

(ii) Using W2.7, or otherwise, deduce that the transfer functions of these linear systems are equal.