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I am interested in results on the eigenvalues of submatrices.

Given a symmetric and positive-semidefinite matrix $M$, denote the submatrix obtained by deleting the ith column and jth row as $[M]_{ji}$.

How does the spectra $\lambda([M]_{ji})$ relate to the spectra $\lambda(M)$?

I know when looking at principal submatrices (ie, $i=j$), we get an interlacing property. However, I can't seem to find such results for other submatrices.

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    $\begingroup$ First, note that your $[M]_{ji}$ is no longer symmetric. Next, note that for the identity matrix $I_n$, $[I_n]_{1n}$ is nilpotent. So I think relatively little can be said. $\endgroup$
    – Willie Wong
    Commented Mar 3, 2011 at 15:06

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Maybe this helps: http://www.math.wm.edu/~ckli/ima/note-2.pdf

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