Parameter-Dependent Rank-One Perturbations of Singular Hermitian Or Symmetric Pencils

Structure-preserving generic low-rank perturbations are studied for classes of structured matrix pencils, including real symmetric, complex symmetric, and complex Hermitian pencils. For singular pencils it is analyzed which characteristic quantities stay invariant in the perturbed canonical form, and it is shown that the regular part of a structured matrix pencil is not affected by generic perturbations of rank one. When the rank-one perturbations involve a scaling parameter, the behavior of the canonical forms dependent on this parameter is analyzed as well.

[1]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[2]  Froilán M. Dopico,et al.  A Note on Generic Kronecker Orbits of Matrix Pencils with Fixed Rank , 2008, SIAM J. Matrix Anal. Appl..

[3]  S. Campbell.  Singular systems of differential equations II , 1980 .

[4]  Nicholas J. Higham,et al.  Detecting the causes of ill-conditioning in structural finite element models , 2014 .

[5]  Volker Mehrmann,et al.  On the distance to singularity via low rank perturbations , 2015 .

[6]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[7]  Froilán M. Dopico,et al.  First order spectral perturbation theory of square singular matrix pencils , 2008 .

[8]  M. Bampton,et al.  Coupling of substructures for dynamic analyses. , 1968 .

[9]  Froilán M. Dopico,et al.  Generic Change of the Partial Multiplicities of Regular Matrix Pencils under Low-Rank Perturbations , 2016, SIAM J. Matrix Anal. Appl..

[10]  R. C. Thompson,et al.  Pencils of complex and real symmetric and skew matrices , 1991 .

[11]  P. Rentrop,et al.  Differential-Algebraic Equations , 2006 .

[12]  Leiba Rodman,et al.  Canonical Forms for Hermitian Matrix Pairs under Strict Equivalence and Congruence , 2005, SIAM Rev..

[13]  Froilán M. Dopico,et al.  Low Rank Perturbation of Kronecker Structures without Full Rank , 2007, SIAM J. Matrix Anal. Appl..

[14]  V. Mehrmann,et al.  Möbius transformations of matrix polynomials , 2015 .

[15]  Froilán M. Dopico,et al.  Low Rank Perturbation of Weierstrass Structure , 2008, SIAM J. Matrix Anal. Appl..

[16]  N. Jacobson,et al.  Basic Algebra I , 1976 .

[17]  Nicola Guglielmi,et al.  On the Nearest Singular Matrix Pencil , 2017, SIAM J. Matrix Anal. Appl..

[18]  V. Mehrmann,et al.  On the sign characteristics of Hermitian matrix polynomials , 2016 .

[19]  C. Mehl.  Anti-triangular and anti-m-Hessenberg forms for Hermitian matrices and pencils , 2000 .

[20]  Volker Mehrmann,et al.  Where is the nearest non-regular pencil? , 1998 .

[21]  Volker Mehrmann,et al.  Numerical methods for parametric model reduction in the simulation of disk brake squeal , 2016 .

[22]  S. Campbell.  Linearization of DAEs along trajectories , 1995 .

[23]  R. C. Thompson,et al.  The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil , 1976 .