Let $\rm A$ be an $m$ by $n$ matrix. You Need to Show that if $\rm A'A$ is idempotent, then $\rm AA'$ is idempotent.

Question

Let $\rm A$ be an $m$ by $n$ matrix. You Need to Show that if $\rm A'A$ is idempotent, then $\rm AA'$ is idempotent.

How should I prove this ?

Answer 1

Here is a guide to how to prove this. Let $M=AA^\top$.

1. Show that $M^3=M^2$
2. Deduce that if $M$ is a diagonal matrix then $M$ is idempotent
3. Reduce to the case that $M$ is a diagonal matrix

You need to use a theorem about real matrices; the result fails for $M=\begin{pmatrix}1\\i\end{pmatrix}\in\mathbb C^2$ for example.