Show that there exists a linear transformation $S\colon V \to V$ such that $S^2 = T$

Question

Let $V$ be a finite-dimensional inner product space over the field $\mathbb{F}$ (which can be $\mathbb{R}$ or $\mathbb{C}$), and let $T\colon V \to V$ be a self-adjoint linear transformation.

If we assume $\mathbb{F} = \mathbb{C}$, how do we show that there exists a linear transformation $S\colon V \to V$ such that $S^2 = T$?

If we assume $\mathbb{F} = \mathbb{R}$ and we suppose that for all $v \in V$, the inner product of $v$ and $T(v)$ is greater than or equal to $0$, how do we show that the conclusion of part(a) still holds about the existence of $S$?

I can't figure out a way to prove these. A little help, please?

Answer 1

Hints:

• If $F = \mathbb C$ and $T$ is diagonal, then you can take a square root of each diagonal entry. Now use the fact that self adjoint linear transformations are diagonalizable.
• If $F = \mathbb R$, then the solution in the previous case cannot be carried out if $T$ has a negative eigenvalue. What does the assumption $\langle v, T(V) \rangle \ge 0$ tell you?