Paper 1, Section I, $2 \mathrm{~B}$

Define the Hermitian conjugate $A^{\dagger}$ of an $n \times n$ complex matrix $A$. State the conditions (i) for $A$ to be Hermitian (ii) for $A$ to be unitary.

In the following, $A, B, C$ and $D$ are $n \times n$ complex matrices and $\mathbf{x}$ is a complex $n$-vector. A matrix $N$ is defined to be normal if $N^{\dagger} N=N N^{\dagger}$.

(a) Let $A$ be nonsingular. Show that $B=A^{-1} A^{\dagger}$ is unitary if and only if $A$ is normal.

(b) Let $C$ be normal. Show that $|C \mathbf{x}|=0$ if and only if $\left|C^{\dagger} \mathbf{x}\right|=0$.

(c) Let $D$ be normal. Deduce from (b) that if $e$ is an eigenvector of $D$ with eigenvalue $\lambda$ then $e$ is also an eigenvector of $D^{\dagger}$ and find the corresponding eigenvalue.

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