Complex symmetric matrix $A$ -- relationship between $AA^*$ and $A^*A$

Question

I've been working on this problem for a while, and haven't been able to figure out where to start from. The problem is:

Let $A$ be a complex symmetric matrix. What is the relationship between $AA^*$ and $A^*A$? Prove your claim.

I've tried using $A = A^T$ and $A^* = \overline{A}^T$ and plugging these definitions into $AA^*$ and $A^*A$, but that hasn't gotten me anywhere significant. Any tips / suggestions / solutions?

Answer 1

Since transpose is related to $^*$, plays well with products, and we're told $A$ is symmetric, it seems sensible to try taking transposes. In doing so, $$ (A A^*)^T = (A \overline{A}^T)^T = \overline{A} A^T = \overline{A} A, $$ and similarly $$ (A^* A)^T = (\overline{A}^T A)^T = A^T \overline{A} = A \overline{A}. $$ These are conjugates of one another, so we see that $$ \overline{(A A^*)^T} = (A^* A)^T, $$ or in other words $$ (A A^*)^* = (A^* A)^T. $$