Matrix version of Pythagoras theorem

I think you are saying: if we know A and B can we find a C such that the equation holds.

A Hermitian Positive Definite matrix M has a Cholesky decomposition $M=CC^{\ast}$. So, if $AA^T+BB^T$ is positive definite symmetric (implying real), then a C exists that satisfies the equation.

A sufficient condition is that A and B are both positive definite.