(a) Give the definition of a normal matrix. Prove that if $A$ is normal, then the (Euclidean) matrix $\ell_{2}$-norm of $A$ is equal to its spectral radius, i.e., $\|A\|_{2}=\rho(A)$.

(b) The advection equation

$$u_{t}=u_{x}, \quad 0 \leqslant x \leqslant 1, \quad 0 \leqslant t<\infty$$

is discretized by the Crank-Nicolson scheme

$$u_{m}^{n+1}-u_{m}^{n}=\frac{1}{4} \mu\left(u_{m+1}^{n+1}-u_{m-1}^{n+1}\right)+\frac{1}{4} \mu\left(u_{m+1}^{n}-u_{m-1}^{n}\right), \quad m=1,2, \ldots, M, \quad n \in \mathbb{Z}_{+}$$

Here, $\mu=\frac{k}{h}$ is the Courant number, with $k=\Delta t, h=\Delta x=\frac{1}{M+1}$, and $u_{m}^{n}$ is an approximation to $u(m h, n k)$.

Using the eigenvalue analysis and carefully justifying each step, determine conditions on $\mu>0$ for which the method is stable. [Hint: All M $\times M$ Toeplitz anti-symmetric tridiagonal (TAT) matrices have the same set of orthogonal eigenvectors, and a TAT matrix with the elements $a_{j, j}=a$ and $a_{j, j+1}=-a_{j, j-1}=b$ has the eigenvalues $\lambda_{k}=a+2 \mathrm{i} b \cos \frac{\pi k}{M+1}$ where $\mathrm{i}=\sqrt{-1}$. ]

(c) Consider the same advection equation for the Cauchy problem $(x \in \mathbb{R}, 0 \leqslant t \leqslant$ $T)$. Now it is discretized by the two-step leapfrog scheme

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

Applying the Fourier technique, find the range of $\mu>0$ for which the method is stable.