Exercise 7.D.15

Proof. The forward direction is obvious, because if $S$ is an isometry, then $\sqrt{S^{\ast }S}$ equals the identity, whose eigenvalues equal $1$.

Suppose all singular values of $S$ equal $1$. This implies that $\sqrt{S^{\ast }S}$, and therefore so does $S^{\text{∗}}S$, equals the identity. 7.42 now implies that $S$ is an isometry. □