Non-zero eigenvalues of $AA^T$ and $A^TA$

Question

As a part of an exercise I have to prove the following:

Let $A$ be an $(n \times m)$ matrix. Let $A^T$ be the transposed matrix of $A$. Then $AA^T$ is an $(n \times n)$ matrix and $A^TA$ is an $(m \times m)$ matrix. $AA^T$ then has a total of $n$ eigenvalues and $A^TA$ has a total of $m$ eigenvalues.

What I need to prove is the following:

$AA^T$ has an eigenvalue $\mu \not = 0$ $\Longleftrightarrow$ $A^TA$ has an eigenvalue $\mu \not = 0$

In other words, they have the same non-zero eigenvalues, and if one has more eigenvalues than the other, then these are all equal to $0$.

How can I prove this?

Thanks and regards.

Answer 1

Let $\lambda$ be an eigenvalue of $A^TA$, i.e. $$A^T A x = \lambda x$$ for some $x \neq 0$. We can multiply $A$ from the left and get $$A A^T (Ax) = \lambda (Ax).$$

What can you conclude from this?

Answer 2

One proof that comes to mind is to use Sylvester's determinant theorem.

In particular: $$ \mu \neq 0 \text{ is an eigenvalue of }A^TA \implies\\ \det(A^TA - \mu I) = 0 \implies\\ \det(I + (-1/\mu)A^TA) = 0 \implies\\ \det(I + A(-1/\mu)A^T) = 0 \implies\\ \det(AA^T - \mu I) = 0 \implies\\ \mu \neq 0 \text{ is an eigenvalue of }AA^T $$

Answer 3

in fact, nonzero eigenvalues $AB$ and $BA$ are the same for any rectangular matrices $A$ and $B$. this follows from the fact that $trace((AB)^k) = trace((BA)^k)$ and the coefficients of the characteristic polynomials of a square matrix $A$ are a function of $trace(A^k).$