Computing stable eigendecompositions of matrix pencils
James Weldon Demmel, Bo Kågström · Linear Algebra and its Applications · 1987
If a matrix pencil A−λB is known only to within a tolerance ϵ (because of measurement or roundoff errors), then it may be difficult to compute a generalized eigendecomposition of A−λB, since its eigenspaces are discontinuous functions of its entries. We are interested in computing an eigendecomposition of A−λB which varies continuously and boundedly as A−λB varies inside a ball of radius ϵ. There are two cases with qualitatively different solutions. The first case is when A−λB is regular, i.e. det(A−λB) is not identically zero. In this case we show how to partition the spectrum of A−λB into disjoint pieces which remain disjoint and whose associated eigenspaces vary smoothly. The second case is when A−λB is singular [i.e. either det(A−λB)≡0 or A−λB is nonsquare]. This case is more difficult than the first, because applications call for computing nongeneric eigenspaces which exist only when A−λB lies in a proper variety (a set of measure zero). The known algorithms for computing these nongeneric structures produce the eigendecomposition of a pencil close to the input which is guaranteed to lie in this proper variety. In this case we prove that as long as the norm of the perturbations produced by the algorithm are smaller than a certain ϵ we can compute from the pencil, the resulting nongeneric eigenspaces and eigenvalues produced by the algorithm vary smoothly. This theorem shows that standard algorithms can compute accurate solutions of many ill-posed problems in systems theory, such as the controllable subspace and uncontrollable modes of a system x̀=Cx+Du.