Given a symmetric nxn matrix matrix and m of the eigenvalue/vector pairs, is there an efficient and numerically stable way to factor out the known structure such that only a smaller (nm)x(nm) problem needs to be solved to get the full set of eigenvalues/vectors ? The known eigenvalues are all zero, so the corresponding eigenvectors span the kernel of the matrix.
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If you already know that all the known eigenvalues of your matrix are zero, it might prove profitable to reduce your original symmetric matrix to tridiagonal form; the nice thing about reducing your matrix to tridiagonal form is that your matrix splits into smaller submatrices if your original matrix had repeated eigenvalues to begin with. 

