Given a symmetric $n\times n$ 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 $(nm)\times (nm)$ 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.

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. 

