Eigenvalues and eigenvectors of a non-symmetric matrix which is a product of 2 symmetric matrices?

Question

I have a non symmetric matrix $AB$ where $A$ and $B$ are symmetric matrices. How can I find the eigenvectors and eigenvalues of $AB$?

In a paper( Fisher Linear Discriminant Analysis by M Welling), the author asks to find eigenvalues and eigenvectors of $B^{1/2} A B^{1/2}$ which is a symmetric matrix. But how can I get eigenvalues and eigenvectors of $AB$ from eigenvalues and eigenvectors of $B^{1/2} A B^{1/2}$?

Could someone please help me?

I tried the solution suggested below. But it doesn't work for me $A$=\begin{pmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{pmatrix} $B$=\begin{pmatrix}32&-12&8\\-12&34&-21\\8&-21&13\end{pmatrix} $B^{1/2}$=\begin{pmatrix}5.53308892146077&-0.950134037741956&0.694386274009086\\-0.950134037741957&4.93157708794602&-2.96256522898147\\0.694386274009086&-2.96256522898147&1.93417552628962\end{pmatrix} eigenvectors of $AB$=\begin{pmatrix}0.516537330395033&-0.781188319935242&-0.0177964973702446\\0.710088559129181&-0.185707982205180&0.521054279012559\\0.478501227273470&0.596034692062496&0.853337988726655\end{pmatrix}

eigenvectors of $B^{1/2} A B^{1/2}$= \begin{pmatrix}0.517933641073670&-0.855373946305353&-0.00895295628145857\\0.725005239437560&-0.444501257554748&0.526104585439372\\0.453995755742558&0.265996323309655&0.850372747536919\end{pmatrix}

$B^{-1/2}$ * (eigenvectors of $B^{1/2} A B^{1/2}$) =\begin{pmatrix}0.0692193234209673&-0.179045453961740&-0.160160577798552\\-0.0951564325356731&-0.0425635779871867&4.68925725410385\\0.0641222410443052&0.136608931923259&7.67966316565345\end{pmatrix}

eigenvectors of $AB$ is different from $B^{-1/2}$ * (eigenvectors of $B^{1/2} A B^{1/2}$). Is there any mistake in what I did?

Answer 1

You simply multiply the eigenvectors with $B^{-1/2}$ (you need that $B$ is positive definite) and the eigenvalues stay the same.

Indeed, having $v$ and eigenvector of $B^{1/2} A B^{1/2}$ to the eigenvalue $\lambda$, i.e., $$ B^{1/2} A B^{1/2} v = \lambda v,$$ you can see easily that with $w= B^{1/2}v$ we have $$ B^{1/2} A B w = B^{1/2} A B^{1/2} v = \lambda v$$ and thus $ A B w = \lambda w$.