About eigenvalues, matrix equation with non invertible matrices

Question

Let A,B be two non-invertible square matrices of the same size.

Would there be a general procedure for solving the equation

$$ Ax = Bkx $$ for both $k$ and $x$? ($k$ is a scalar).

For example, if B were invertible, we could rewrite the equation as $$B^{-1}Ax = kx$$ and then we could just solve for the eigenvalues and eigenvectors of $B^{-1}A$. Is there any way to approach this problem if $A$ and $B$ are not invertible?

Answer 1

If you consider $x$ in the kernel of $B$, then $Bkx$ will be zero; unless $x$ is also in the kernel of $A$, this cannot be a solution. So solutions come in two classes:

• Those in $Ker(B) \cap Ker(A)$, and
• Those not in $Ker(B)$.

For those in the second group, you can take the Moore-Penrose pseudo-inverse $H$ of $B$, and for these, $HBx = x$, so you can write $$ (HA)x = k(HB)x = kx $$ and merely look for eigenvectors of the matrix $HA$.