# What is meant by an eigenvalue of 2 matrices?

In looking for a way to compare covariance matrices, I came across a paper that formulates a metric using what appears to be a joint eigenvalue. I'm not familiar with this idea.

Thus we propose the distance metric

$$d(A,B) = \sqrt{\sum^n_{i=1}ln^2\lambda_i(A,B)}$$

between symmetric positive semidefinite matrices $$A$$ and $$B$$, with the eigenvalues $$\lambda_i(A,B)$$ from $$|\lambda A - B| = 0$$."

I imagine by hand this is calculated essentially the same way as the eigenvalues and eigenvectors of a single matrix that I'm familiar with. But as I will be using this in a computer program, I need to understand what its talking about in order to find a numerical solution.

EDIT: In another paper (the one I'm primarily using), they present this as the generalize eigenvalues and eigenvectors computed from:

$$\lambda_i C_1 x_i − C_2 x_i = 0$$, $$for$$ $$i = 1...d$$

Where $$C_1$$ and $$C_2$$ are covariance matrices and $$x_i$$ are the eigenvectors.

• This evidently generalizes the classical notion of eigenthings. Taking $A=I$, you recover the classical equation.
– MPW
Apr 23 '15 at 2:12