# Geometric Multiplicity, Eigenspace of Matrix

If $A$ is an $r$ x $s$ matrix and B is an $s$ x $r$ matrix.

$E_\mu(C)$ is the eigenspace of square matrix C with eigenvalue $\mu ≠ 0$.

Proof:

$dimE_\mu(AB) = dimE_\mu(BA)$

I imagine one would need to find the basis of eigenvectors for both spaces, but I'm not sure how...

Hint: If $v$ is an eigenvector for $AB$, then $Bv$ is an ...
• Eigenvector of $BA$. Does that imply that the spaces have the same number of eigenvectors? – user79572 May 26 '13 at 19:20