self-adjoint operator and unitary orthogonal matrix

Question

Please offer a solution to the following problem. It was offered in class by my professor as an additional exercise to try on one's own.

Let $V$ be the inner product space, and assume that $\alpha \in End(V)$. Suppose that $A$ is the representation matrix of $\alpha$ with respect to an orthonormal basis {$v_1,...,v_n$}.

(i) Prove that $\alpha$ is self-adjoint if and only if $A = A^*$.

(ii) Prove that if $B$ is the representation matrix of $\alpha$ with respect to another orthonormal basis {$w_1,...,w_n$}, then $B=U^*AU$ for some matrix $U$ such that $U^*U=I$.

Thank you for your assistance.

Answer 1

For part 1, use Proposition 16.16 from Golan.

Let $V$ and $W$ be finitely-generated inner product spaces, having ONB $B=\{v_1, \ldots, v_n\}$ and $D=\{w_1, \ldots, w_n\}$, respectively. Let $\alpha: V \to W$ be a linear transformation. Then $\Phi_{BD}(\alpha)$ is the matrix $A=[a_{ij}]$, where $a_{ji}=\langle \alpha(v_i),w_j\rangle$ and $\Phi_{DB}(\alpha^*)=A^H$.

The proof of this proposition follows: For all $1 \leq i \leq n$, let $\alpha(v_i)= \displaystyle\sum_{h=1}^k a_{hj} w_h$. Then for all $1 \leq j \leq k$, we have $\langle \alpha(v_i), w_j \rangle= \langle \sum_{h=1}^k a_{hj} w_h, w_j \rangle=a_{ji}$ and also $\langle \alpha^*(w_j),v_i \rangle= \overline{\langle v_i, \alpha^*(w_j) \rangle}= \overline{\langle \alpha(v_i), w_j \rangle}=\overline{a_{ji}}$ as needed.

The proof for this proposition is similar to the proof of Part 1.