1
$\begingroup$

Let $P ∈ \text{Mat}_{n\times m}(\Bbb{R}), Q ∈ \text{Mat}_{m\times n}(\Bbb{R})$ .
Can someone explain to me how to show that if $I + P Q$ is nonsingular then $I + QP$ is also nonsingular?

Nonsingularity means that if $PQ$ exists, then $QP$ is equivalent to that.. So how do I formulate this neatly? It seems pretty obvious to me?

$\endgroup$
3
$\begingroup$

The matrix $I+PQ$ is nonsingular if and only if $-1$ is not an eigenvalue of $PQ$.

Now prove that $PQ$ and $QP$ have the same nonzero eigenvalues.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.