Prove that all Toeplitz tridiagonal $M \times M$ matrices $A$ of the form

$$A=\left[\begin{array}{rrrrr} a & b & & & \\ -b & a & b & & \\ & \ddots & \ddots & \ddots & \\ & & -b & a & b \\ & & & -b & a \end{array}\right]$$

share the same eigenvectors $\left(\boldsymbol{v}^{(k)}\right)_{k=1}^{M}$, with the components $\boldsymbol{v}_{m}^{(k)}=i^{m} \sin \frac{k m \pi}{M+1}, m=$ $1, \ldots, M$, where $i=\sqrt{-1}$, and find their eigenvalues.

The advection equation

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

is approximated by the Crank-Nicolson scheme

$$u_{m}^{n+1}-u_{m}^{n}=\frac{1}{4} \mu\left(u_{m+1}^{n+1}-u_{m-1}^{n+1}\right)+\frac{1}{4} \mu\left(u_{m+1}^{n}-u_{m-1}^{n}\right)$$

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

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

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