Prove that a positive operator is invertible Help guys, I need to prove this:
Let $(V,\langle~,~ \rangle)$ a finite n-dimensional euclidean space.Let $T$ be a linear operator defined positive (There exists a non singular operator $S$ such that $T=S^*S$) on $V$, prove that $T$ is invertible
I tried this:
We know by hypothesis that $S$ is ivertible,then there exists $S^{-1}$ such that $SS^{-1}=I$ $S$ then I need to prove that $T=T^*$
But that is trivial, since $T=S^*S=(S^*S)^*=S^*(S^*)^*=S^*S=T$
Is that correct? I need help!

In addition to what others have said, you could just use a determinant argument
$$\det T = \det S^* \cdot \det S = \overline{\det S} \det S = |\det S|^2 > 0$$
Though as others have said, technically this is a positive-definite operator.
