how to prove that $BB^H$ must be invertible.If $\mathbf B$ is a $N$ by $M$ matrix,and $N<M$

Question

If $\mathbf B$ is a $N$ by $M$ matrix,and $N<M$.how to prove that $BB^H$ must be invertible.

Here we assume that $N$ is the rank(B)

Answer 1

$B B^H$ is always Hermitian and positive semidefinite. Note that $x^H B B^H x = \|B^H x\|^2$, so $x^H B B^H x = 0$ implies $B^H x = 0$. If $B$ (and thus $B^H$) has rank $N$, this can only happen for $x=0$, so $B B^H$ is positive definite and therefore invertible.