Find eigenvalues and eigenvectors: strange case Why is this expression:
$$\begin{pmatrix}
\frac{k+mg}{l} & -k\\ 
-k & \frac{k+mg}{l}
\end{pmatrix} \begin{pmatrix}
\rho_1\\\rho_2 
\end{pmatrix}=\omega^2\begin{pmatrix}
m & 0\\ 
0 & m
\end{pmatrix} \begin{pmatrix}
\rho_1\\\rho_2 
\end{pmatrix}$$
a matricial form for equation for eigenvalues and eigenvectors? I was told that the generical expression of the equation for eigenvalues and eigenvectors is $A\bf{x}=\lambda \bf{x}$... How can I obtain eigenvalues and eigenvectors from the first expression that I have written?
Thank you!

If $A$ is the big matrix on the left and $x = (\rho_1, \rho_2)^T$, then the equation you have written can be rewritten as
$$ Ax = \omega^2 m I x$$
where $I$ is the identity matrix. But this can thus be written as $Ax = \omega^2 m x$, so this is just the eigenvalue-eigenvector equation with eigenvalue $\lambda = \omega^2 m$.
