×

Rank one perturbations of \(H\)-positive real matrices. (English) Zbl 1283.15102

Summary: We consider a generic rank one structured perturbation on \(H\)-positive real matrices. The case with a complex rank one perturbation is treated in general, but the main focus of this article is the real rank one perturbation. In general, the \(H\)-positive real matrix \(A\) which is given in Jordan canonical form looses the largest Jordan block after a rank one perturbation for each eigenvalue. Surprisingly, for a real \(H\)-skew symmetric matrix for which the largest Jordan block at eigenvalue zero has even size and for a real \(H\)-nonnegative rank one perturbation the largest Jordan block with zero eigenvalue grows one in size. Generic Jordan structures of perturbed matrices are identified.

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
15A03 Vector spaces, linear dependence, rank, lineability
15A21 Canonical forms, reductions, classification
15B57 Hermitian, skew-Hermitian, and related matrices
15A18 Eigenvalues, singular values, and eigenvectors
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dopico, F.; Moro, J., Low rank perturbation of Jordan structure, SIAM J. Matrix Anal. Appl., 25, 495-506 (2003) · Zbl 1055.15017
[3] Hörmander, L.; Melin, A., A remark on perturbations of compact operators, Math. Scand., 75, 255-262 (1994) · Zbl 0824.47008
[4] Mehl, Chr.; Mehrmann, V.; Ran, A. C.M.; Rodman, L., Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations, Linear Algebra Appl., 435, 687-716 (2011) · Zbl 1223.15015
[6] Lancaster, P.; Rodman, L., Cannonical forms for symmetric/skew-symmetric real matrix pairs under strict equivalence and congruence, Linear Algebra Appl., 406, 1-76 (2005) · Zbl 1081.15007
[7] Lancaster, P.; Tismenetsky, M., The Theory of Matrices (1985), Academic Press: Academic Press Orlando · Zbl 0516.15018
[8] Ran, A. C.M.; Wojtylak, M., Eigenvalues of rank one perturbations of unstructured matrices, Linear Algebra Appl., 437, 589-600 (2012) · Zbl 1247.15009
[9] Rodman, L., Similarity vs unitary similarity and perturbation analysis of sign characteristics: complex and real indefinite inner products, Linear Algebra Appl., 416, 945-1009 (2006) · Zbl 1098.15023
[10] Savchenko, S. V., On a generic change in the spectral properties under perturbation by an operator of rank one, Mat. Zametki, 74, 4, 590-602 (2003), translation in Math. Notes 74 (3-4) (2003) 557-568
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.