I was just thinking about this fact. It's pretty straightforward to see if we invoke the spectral theorem. But I was curious if someone knew a way to do it way to do it without the spectral theorem.
I also considered using the fact that $A$ has a square root. That is $A = R^*R$ for some matrix $R$ but the only way I know how to prove this fact invokes the spectral theorem so that doesn't really count. So I guess another way to prove this that doesn't use the spectral theorem would also answer my question.