Computing extreme subspaces using Mirsky theorem

Extreme eigenpairs computation is of considerable interest in signal processing and estimation. Thus problem of simultaneous computation of the smallest and largest eigenvalues and the corresponding eigenvectors of a symmetric matrix is considered. The proposed methods are derived from optimizing cost functions which are chosen to have optimal values at vectors that are linear combinations of extreme eigenvectors of a given matrix. Dynamical systems that converge to extreme eigenvectors are derived from necessary optimality conditions which are given in terms of a gradient of certain cost functions over a Stiefel manifold. Numerical examples are given to examine the convergence.