# eigenvectors and eigenvalues problem

I have solved an eigenvalue problem for matrix A which is orthogonal. I am trying to prove that the eigenvectors for matrix B is the same and find its eigenvalues. Matrix B is related to A as follows...

All I know is that the eigenvalues of A^-1 are the inverse of eigenvalues of A. please help!

• So is $x$ any column of $B$? – Sean Roberson May 30 '17 at 20:58
• sean i have updated – gamma1 May 30 '17 at 21:04

Based on my interpretation, here's how it goes.

Let $(\lambda, y)$ be an eigenpair of $A$ (that is, $Ay = \lambda y$). Then

\begin{align*} By &= Ay + A^{-1} y + xIy \\ &= \lambda y + \frac{1}{\lambda} y + xy \\ &= \left( \lambda + \frac{1}{\lambda} + x \right) y \end{align*}

and so $\left( \lambda + \frac{1}{\lambda} + x, y \right)$ is an eigenpair for $B$.

• ah Ok, so y is your eigenvector. you initially state that it is an eigenvector of A, let B apply to it, and see what comes out? – gamma1 May 30 '17 at 21:29
• Yeah, pretty much. – Sean Roberson May 30 '17 at 21:30