Exercise 6.5.18

Write $A\sim B$ to say that $A$ is unitarily equivalent to $B$. Check the three conditions in the Appendix A.

reflexivity

Since $A=I^{\text{∗}}\mathit{AI}$, we get $A\sim A$.

symmetry

If $A\sim B$, we have $B=P^{\text{∗}}\mathit{AP}$ and so $A=\mathit{PB}{P}^{\text{∗}}={(P^{\text{∗}})}^{\text{∗}}B{P}^{\text{∗}}$. This means that $B\sim A$.

transitivity

If $A\sim B$ and $B\sim C$, we have $B=P^{\text{∗}}\mathit{AP}$ and $C=Q^{\text{∗}}\mathit{BQ}$. This means that

$$C=Q^{\text{∗}}\mathit{BQ}=Q^{\text{∗}}{P}^{\text{∗}}\mathit{APQ}={(\mathit{PQ})}^{\text{∗}}A(\mathit{PQ})$$

and so $A\sim C$.