Exercise 6.5.7

By the Corollary 2 after Theorem 6.18, we may find an orthonormal basis $\beta$ such that

Also, since the eigenvalue ${\lambda }_i$ has its absolute value $1$, we may find some number ${\mu }_i$ such that ${\mu }_i^2={\lambda }_i$ and $\left|{\mu }_i\right|=1$. Denote

to be an unitary operator. Now pick $U$ to be the matrix whose matrix representation with respect to $\beta$ is $D$. Thus $U$ is unitary and $U^2=T$.