Is $A^*A$ a positive definite matrix?

Question

Given $A \in \mathbb{C}^{n \times n}$, where $A^*$ is the conjugate transpose, I want to know if the matrix $A^* A$ is a positive definite. Recall that a matrix $B$ is positive define if and only if, $$v^*Bv >0$$ for all $v\in \mathbb{C}^{n} \setminus \{ 0 \}$.

Answer 1

It depends on wheher $A$ is singular. If $A$ is non-singular, then $Av$ will be non-zero for non-zero $v$ and $$v^*A^*Av=(Av)*(Av)>0$$.If $A$ is singular, hen $Av=0$for some non-zero $v$, so $v^*A^*Av=0$.