# Difference in eigenvalue equations

Mathematics Asked on January 1, 2022

I came across a strange equation in the solution to a problem. It looks like this:$$My – Mx = lambda x$$

In this problem $$M$$ is an $$ntimes n$$ (full rank) matrix, and $$x$$ and $$y$$ are vectors. ($$y$$ is given in the problem and I am looking for $$x$$). Is there a solution for $$x$$ here?

Suppose that $$M+lambda$$ is invertible. Then $$My = (M+lambda)x$$ which means that $$(M+lambda)^{-1} M y = x$$ which would then directly give you $$x$$ if you know $$y,lambda, M$$.

Answered by Frederik Ravn Klausen on January 1, 2022

Rearrange the equation as $$y=(lambda M^{-1}+I)x$$. Therefore it is solvable if and only if $$y$$ lies inside the range of $$lambda M^{-1}+I$$, i.e. if and only if $$operatorname{rank}(lambda M^{-1}+I)=operatorname{rank}pmatrix{lambda M^{-1}+I&y}$$.

Answered by user1551 on January 1, 2022

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