# Eigenvalues of submatrices

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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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. – Willie Wong Mar 3 '11 at 15:06