# Paper 1, Section II, $7 \mathrm{~A}$

Let $A$ be an $n \times n$ Hermitian matrix. Show that all the eigenvalues of $A$ are real.

Suppose now that $A$ has $n$ distinct eigenvalues.

(a) Show that the eigenvectors of $A$ are orthogonal.

(b) Define the characteristic polynomial $P_{A}(t)$ of $A$. Let

$P_{A}(t)=\sum_{r=0}^{n} a_{r} t^{r}$

Prove the matrix identity

$\sum_{r=0}^{n} a_{r} A^{r}=0$

(c) What is the range of possible values of

$\frac{\mathbf{x}^{\dagger} A \mathbf{x}}{\mathbf{x}^{\dagger} \mathbf{x}}$

for non-zero vectors $\mathbf{x} \in \mathbb{C}^{n} ?$ Justify your answer.

(d) For any (not necessarily symmetric) real $2 \times 2$ matrix $B$ with real eigenvalues, let $\lambda_{\max }(B)$ denote its maximum eigenvalue. Is it possible to find a constant $C$ such that

$\frac{\mathbf{x}^{\dagger} B \mathbf{x}}{\mathbf{x}^{\dagger} \mathbf{x}} \leqslant C \lambda_{\max }(B)$

for all non-zero vectors $\mathbf{x} \in \mathbb{R}^{2}$ and all such matrices $B$ ? Justify your answer.