I am working on a paper and I have a complex-valued $N \times K$ matrix $A$ and its complex conjugate transpose $A^*$. What happens when I take a product $A A^*$? What does the property state what happens when I multiply them other than having a real-valued matrix?
Here N>>K
Best Answer
It may be instructive to look at the SVD of $A$. Let
$$ A = U \begin{bmatrix} \Sigma\\ 0 \end{bmatrix} V^*, $$
where $\Sigma$ is the $K\times K$ diagonal matrix of singular values. Then
$$ AA^* = U \begin{bmatrix} \Sigma^2 & 0\\ 0 & 0 \end{bmatrix} U^* = \sum_{i=1}^r \sigma_i^2 u_iu_i^*, $$ where $r \leq K$ is the rank of $A$. Thus, the rank of $AA^*$ is equal to the rank of $A$, its column space is the span of $U_{1:r}$ (the same as $A$), and its nullspace is the span of $U_{r+1:N}$.
I am not sure if this is what you were looking for but I hope it helps.