(a) Prove that all Toeplitz symmetric tridiagonal $M \times M$ matrices

$$A=\left[\begin{array}{ccccc} a & b & 0 & \cdots & 0 \\ b & a & b & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots & b & a & b \\ 0 & \cdots & 0 & b & a \end{array}\right]$$

share the same eigenvectors $\left(v^{(k)}\right)_{k=1}^{M}$ with components $v_{i}^{(k)}=\sin \frac{k i \pi}{M+1}$, $i=1, \ldots, M$, and eigenvalues to be determined.

(b) The diffusion equation

$$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}, \quad 0 \leqslant x \leqslant 1, \quad 0 \leqslant t \leqslant T$$

is approximated by the Crank-Nicolson scheme

$$\begin{aligned} u_{m}^{n+1}-\frac{1}{2} \mu\left(u_{m-1}^{n+1}-2 u_{m}^{n+1}+u_{m+1}^{n+1}\right) &=u_{m}^{n}+\frac{1}{2} \mu\left(u_{m-1}^{n}-2 u_{m}^{n}+u_{m+1}^{n}\right), \\ \text { for } \quad m &=1, \ldots, M, \end{aligned}$$

where $\mu=\Delta t /(\Delta x)^{2}, \quad \Delta x=1 /(M+1)$, and $u_{m}^{n}$ is an approximation to $u(m \Delta x, n \Delta t)$. Assuming that $u(0, t)=u(1, t)=0, \forall t, \quad$ show that the above scheme can be written in the form

$$B u^{n+1}=C u^{n}, \quad 0 \leqslant n \leqslant(T / \Delta t)-1$$

where $u^{n}=\left[u_{1}^{n}, \ldots, u_{M}^{n}\right]^{\top}$ and the real matrices $B$ and $C$ should be found. Using matrix analysis, find the range of $\mu$ for which the scheme is stable. [Do not use Fourier analysis.]