Paper 1, Section II, 7 A7 \mathrm{~A}

Vectors and Matrices | Part IA, 2009

Let AA be an n×nn \times n Hermitian matrix. Show that all the eigenvalues of AA are real.

Suppose now that AA has nn distinct eigenvalues.

(a) Show that the eigenvectors of AA are orthogonal.

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

PA(t)=r=0nartrP_{A}(t)=\sum_{r=0}^{n} a_{r} t^{r}

Prove the matrix identity

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

(c) What is the range of possible values of

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

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

(d) For any (not necessarily symmetric) real 2×22 \times 2 matrix BB with real eigenvalues, let λmax(B)\lambda_{\max }(B) denote its maximum eigenvalue. Is it possible to find a constant CC such that

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

for all non-zero vectors xR2\mathbf{x} \in \mathbb{R}^{2} and all such matrices BB ? Justify your answer.

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