On a Perturbation Bound for Invariant Subspaces of Matrices

Given a nonsymmetric matrix $A$, we investigate the effect of perturbations on an invariant subspace of $A$. The result derived in this paper differs from Stewart's classical result and sometimes yields tighter bounds. Moreover, we provide norm estimates for the remainder terms in well-known perturbation expansions for invariant subspaces, eigenvectors, and eigenvalues.

[1]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[2]  J. Demmel.  Computing stable eigendecompositions of matrices , 1986 .

[3]  Ming Gu,et al.  An Algorithm to Compute Sepλ , 2006, SIAM J. Matrix Anal. Appl..

[4]  E. Zeidler.  The Implicit Function Theorem , 1995 .

[5]  G. Stewart.  Error Bounds for Approximate Invariant Subspaces of Closed Linear Operators , 1971 .

[6]  Ji-guang Sun,et al.  Perturbation expansions for invariant subspaces , 1991 .

[8]  W. Kahan,et al.  The Rotation of Eigenvectors by a Perturbation. III , 1970 .

[9]  On ɛ-spectra and stability radii , 2002 .

[10]  C. D. Meyer,et al.  Derivatives and perturbations of eigenvectors , 1988 .

[11]  V. N. Bogaevski,et al.  Matrix Perturbation Theory , 1991 .

[12]  Lower bound estimation for the separation of two matrices , 1997 .

[13]  P. Lancaster,et al.  Invariant subspaces of matrices with applications , 1986 .

[14]  F. Chatelin.  Spectral approximation of linear operators , 2011 .

[15]  J. Varah.  On the Separation of Two Matrices , 1979 .

[16]  Raúl Tempone,et al.  On NonAsymptotic Optimal Stopping Criteria in Monte Carlo Simulations , 2014, SIAM J. Sci. Comput..

[17]  R. Byers.  A LINPACK-style condition estimator for the equation AX-XB^{T} = C , 1984 .

[18]  Chandler Davis.  The rotation of eigenvectors by a perturbation , 1963 .

[19]  Daniel Kressner,et al.  An Error Analysis of Galerkin Projection Methods for Linear Systems with Tensor Product Structure , 2013, SIAM J. Numer. Anal..

[20]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[21]  Daniel Kressner,et al.  Numerical Methods for General and Structured Eigenvalue Problems , 2005, Lecture Notes in Computational Science and Engineering.

[22]  P. Lancaster,et al.  Invariant Subspaces of Matrices with Applications (Classics in Applied Mathematics, 51) , 2006 .

[23]  L. Trefethen.  Spectra and pseudospectra , 2005 .

[24]  G. Stewart.  Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems , 1973 .

[25]  R. Alam,et al.  On Stable Eigendecompositions of Matrices , 2005, SIAM J. Matrix Anal. Appl..

[26]  James Demmel,et al.  On computing condition numbers for the nonsymmetric eigenproblem , 1993, TOMS.