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 $operatorname{rank}(lambda M^{-1}+I)=operatorname{rank}pmatrix{lambda M^{-1}+I&y}$.

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