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.

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

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