Perturbation analysis of singular subspaces and deflating subspaces

Abstract

Perturbation expansions for singular subspaces of a matrix and for deflating subspaces of a regular matrix pair are derived by using a technique previously described by the author. The perturbation expansions are then used to derive Fréchet derivatives, condition numbers, and rth-order perturbation bounds for the subspaces. Vaccaro's result on second-order perturbation expansions for a special class of singular subspaces can be obtained from a general result of this paper. Besides, new perturbation bounds for singular subspaces and deflating subspaces are derived by applying a general theorem on solution of a system of nonlinear equations. The results of this paper reveal an important fact: Each singular subspace and each deflating subspace have individual perturbation bounds and individual condition numbers. Mathematics Subject Classification (1991): 65F35 PreliminariesAlthough perturbation bounds for subspaces associated with certain matrix eigenvalue problems and the singular value decomposition of a matrix have been investigated by numerous authors (see, e.g., [1, 4-6, 8-10, 15-18, 21, 23-25]), relatively little attention has been paid to perturbation expansions for the subspaces. Kato [9] gives perturbation series for projections corresponding to invariant subspaces. The author [20] gives perturbation expansions for invariant subspaces. Recently, Vaccaro [22] gives a second-order perturbation expansion for a special class of singular subspaces.

Perturbation analysis for the sign functions of regular matrix pairs

“…If (A 11 , B 11 )∩ (A 22 , B 22 ) = ∅, then W and L are nonsingular (see [17]). In this case, it is easy to verify that…”

An Arithmetic for Matrix Pencils: Theory and New Algorithms

“…In the table, the estimate is denoted by ε/dif. See [5,47] for details. It demonstrates how ill-conditioned E and/or j in (41) can adversely affect both forward and The tables compares (41), the Q Z algorithm [2,41] and Algorithm 1.…”

“…In the right-most-column, ε/dif is an estimate of the largest forward error that can be caused by a backward error of roughly the unit round. See [5,47] for details backward errors. In contrast, the backward numerically stable Q Z algorithm is unaffected.…”

“…This and the constraint α/β = 1/2 establish α = ±1/ √ 2, β = ± √ 2. The (1, 2) block of (46) now follows from (47). The (2, 2) block of (46) follows from (50).…”

Perturbation analysis of generalized singular subspaces

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This paper, as a continuation of the paper [20] in Numerische Mathematik, studies the subspaces associated with the generalized singular value decomposition. Second order perturbation expansions, Fréchet derivatives and condition numbers, and perturbation bounds for the subspaces are derived.

Mathematics Subject Classification (1991): 65F15

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