MATH319 Exercises 4 Workshop Exercise 2 6 Workshop Exercises 3

5 Assessed Exercise 2

A2.1 Let

A=\left[\begin{pmatrix}1&-1\cr 3&5\cr\end{pmatrix}\right];

find an invertible matrix $S$ and a diagonal matrix $D$ such that

A=SDS^{-1}.

Hence or otherwise find $\exp(tA)$ , where $t$ is a real variable.

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A2.2 Find a matrix $A$ such that $s^{4}+2s^{3}+s^{2}+4s+2$ is the characteristic polynomial of $A$ . Then find the eigenvalues of $A$ numerically.

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A2.3 Let $A$ be a $(n\times n)$ complex matrix, and for $C$ a $(1\times n)$ row vector let

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

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

(ii) Show that $W$ has dimension one, if and only if $C^{T}$ is an eigenvector of $A^{T}$ . Here $A^{T}$ is the transpose of $A$ .

(iii) Use the Cayley–Hamilton theorem to show that, for all $A$ , right multiplication $R_{A}:X\mapsto XA$ for $X\in W$ defines a linear operator $R_{A}:W\rightarrow W$ .

(iv) Given that dimension of $W$ equals the rank of $P$ , where

P=\left[\begin{pmatrix}C\cr CA\cr\vdots\cr CA^{n-1}\cr\end{pmatrix}\right],

calculate the dimension of $W$ 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 C=\left[\begin{pmatrix}1&-5&-1/2&3\cr\end{pmatrix}\right].

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A2.4 Let $T(s)$ be a complex rational function, not necessarily proper. Show that:

(i) $T(s)$ is proper, if and only if there exists $L\in{\bf C}$ such that $T(s)\rightarrow L$ as $|s|\rightarrow\infty$ ;

(ii) $T(s)$ is strictly proper, if and only if $T(s)\rightarrow 0$ as $|s|\rightarrow\infty$ .

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