MATH319 Exercises 1 Instructions concerning submitted coursework 3 Assessed Exercise 1

2 Workshop Exercise 1

W1.3 A complex square matrix is stable if all the eigenvalues $\lambda$ have $\Re\lambda<0$ , where $\Re\lambda$ is the real part of $\lambda$ . For each of the following matrices, find the eigenvalues numerically using computer software to test whether $\pm A,\pm B,\pm C$ are stable:

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

W1.4 Let

A=\left[\begin{pmatrix}1&4&5\cr 6&2&1\cr 1&7&8\cr\end{pmatrix}\right],\quad I=% \left[\begin{pmatrix}1&0&0\cr 0&1&0\cr 0&0&1\cr\end{pmatrix}\right],

B=\left[\begin{pmatrix}1&2\cr 7&4\cr 5&2\cr\end{pmatrix}\right],\quad C=\left[% \begin{pmatrix}0&2&7\cr 1&4&3\cr\end{pmatrix}\right],\quad D=\left[\begin{% pmatrix}3&5\cr 6&7\cr\end{pmatrix}\right].

Compute the matrix transfer function

T(s)=C(sI-A)^{-1}B+D

either by hand or using suitable computer software. Here $s$ is an algebraic variable.

W1.5 A coupled harmonic oscillator satisfies the system of differential equations with $t$ time, $k$ and $c$ constants, $u_{1}$ and $u_{2}$ inputs and $y_{1}$ and $y_{2}$ state variables, namely

{{d^{2}y_{1}}\over{dt^{2}}}=-ky_{1}+c(y_{2}-y_{1})+u_{1}

and

{{d^{2}y_{2}}\over{dt^{2}}}=-ky_{2}-c(y_{2}-y_{1})+u_{2}.

By introducing extra state variables $v_{1}=dy_{1}/dt$ and $v_{2}=dy_{2}/dt,$ write this as a system of first order differential equations, in matrix form.

W1.6 Let

A=\left[\begin{pmatrix}0&1&0&0\cr 3&0&0&2\cr 0&0&0&1\cr 0&-2&0&1\cr\end{% pmatrix}\right],\quad B=\left[\begin{pmatrix}0&0\cr 1&0\cr 0&0\cr 0&1\cr\end{% pmatrix}\right];

let $Q$ be the $(4\times 8)$ matrix written as $(4\times 2)$ blocks

Q=\left[\begin{pmatrix}B&AB&A^{2}B&A^{3}B\cr\end{pmatrix}\right].

Find the rank of $Q$ .