A question about a linear algebra proof

A question about a linear algebra proof

Consider following statement:
Every positive operator $T:V\to V$ where $V$ is a finite dimensional
complex innner product space has a unique positive square root.
The proof in my notes is this: use previous theorems to write $V$ as
direct sum of eigen spaces for eigenvalues of $T$, to get existence of
square root $S$ and to write $V$ as direct sum of eigen spaces of $S$.
Etc.
It is not so long. But I wonder if it could be proved like this: (can you
tell me if following is a valid proof?)
On complex vector spaces $V$ every operator $T$ has an eigen basis.
Represented in the eigen basis $T$ is a diagonal matrix $D$. Then
$S=\sqrt{D}$ is a square root of $T$. By assumption $T$ is positive and
therefore has only non negative real eigen values. Therefore $S$ is a
positive square root.
Please can you tell me if this is a valid proof of the theorem?