Mehl, Christian

Mehrmann, Volker

Wojtylak, Michał

Operators and Matrices

Christian Mehl

Volker Mehrmann

Michał Wojtylak

Crossref

On the distance to singularity via low rank perturbations

Jagiellonian Univeristy Repository

OperatorsandMatricesVolume 9, Number 4 (2015), 733–772 doi:10.7153/oam-09-44ON THE DISTANCE TO SINGULARITYVIA LOW RANK PERTURBATIONSCHRISTIAN MEHL, VOLKER MEHRMANN AND MICHAŁ WOJTYLAKAbstract. For regular matrix pencils the distance in norm to the nearest singular pencil under lowrank perturbation is studied. Characterizations of this distance are derived via the Weyl functionof the perturbation. Special attention is paid to the Hermitian pencil case. Estimates for thedistance of a given pencil to the set of singular pencils are obtained.Mathematics subject classification (2010): 15A18, 15A22, 47A75, 65F15, 65F22.Keywords and phrases: Low rank perturbation, singular pencils, distance to singularity, Weyl function.RE F ER EN C ES[1] H. ATTOUCH, J. BOLTE, P. REDONT, AND A. SOUBEYRAN, Proximal alternating minimization andprojection methods for nonconvex problems: an approach based on the Kurdyka-łojasiewicz inequalityMath. Oper. Res., 35: 438–457, 2010.[2] K. E. BRENAN, S. L. CAMPBELL, AND L. R. PETZOLD, Numerical Solution of Initial-Value Prob-lems in Differential Algebraic Equations, SIAM Publications, Philadelphia, PA, 2nd edition, 1996.[3] R. BYERS, C. HE, AND V. MEHRMANN, Where is the nearest non-regular pencil, Linear AlgebraAppl., 285: 81–105, 1998.[4] F. DE TERA´N AND F. DOPICO, Low rank perturbation of regular matrix polynomials, Linear AlgebraAppl., 430: 579–586, 2009.[5] F. DE TERA´N AND F. DOPICO, First order spectral perturbation theory of square singular matrixpolynomials, Linear Algebra Appl., 432: 892–910, 2010.[6] F. DE TERA´N, F. DOPICO, AND J. MORO, First order spectral perturbation theory of square singularmatrix pencils, Linear Algebra Appl., 429: 548–576, 2008.[7] E. EICH-SOELLNER AND C. FU¨HRER, Numerical Methods in Multibody Systems, B. G. TeubnerStuttgart, 1998.[8] F. R. GANTMACHER, Theory of Matrices, Chelsea, New York, 1959.[9] I. GOHBERG, P. LANCASTER, AND L. RODMAN, Indefinite Linear Algebra and Applications,Birkha¨user, Basel, 2005.[10] G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, Johns Hopkins University Press, Balti-more, 3rd edition, 1996.[11] N. J. HIGHAM, D. S. MACKEY, N. MACKEY, AND F. TISSEUR, Symmetric linearizations for matrixpolynomials, SIAM J. Matrix Anal. Appl., 29 (1): 143–159, 2006.[12] T. KATO, Perturbation Theory for Linear Operators, Springer Verlag, New York, N.Y., 1966.[13] S. KRANTZ AND H. PARKS, A Primer of Real Analytic Functions, Birkha¨user, Boston, MA, 2ndedition, 2002.[14] P. KUNKEL AND V. MEHRMANN, Differential-Algebraic Equations. Analysis and Numerical Solu-tion, EMS Publishing House, Zu¨rich, Switzerland, 2006.[15] R. LAMOUR, R. MA¨RZ, AND C. TISCHENDORF, Differential-Algebraic Equations: A ProjectorBased Analysis, Differential-Algebraic Equations Forum, Springer Verlag, 2013.[16] D. MACKEY, N. MACKEY, C. MEHL, AND V. MEHRMANN, Structured polynomial eigenvalue prob-lems: Good vibrations from good linearizations, SIAM J. Matrix Anal. Appl., 28 (4): 1029–1051,2006.c©    , ZagrebPaper OaM-09-44734 C. MEHL, V. MEHRMANN AND M. WOJTYLAK[17] The MathWorks, Inc., Natick, MA, Simscape, 2013., http://www.mathworks.de/products/simscape/.[18] The MathWorks, Inc., Natick, MA, Simulink, 2013., http://www.mathworks.de/products/simulink/.[19] MATLAB, Version r2013b, The MathWorks, inc., 24 Prime Park Way, Natick, MA 01760-1500, USA,2013.[20] Modelica Association, Modelica standard library 3.2.1., https://www.modelica.org/.[21] A. C. M. RAN AND M. WOJTYLAK, Eigenvalues of rank one perturbations of unstructured matrices,Linear Algebra Appl., 437: 589–600, 2012.[22] R. RIAZA, Differential-algebraic systems. Analytical aspects and circuit applications, World Scien-tific Publishing Co. Pte. Ltd., Hackensack, NJ., 2008.[23] S. SAVCHENKO, On the Perron roots of principal submatrices of co-order one of irreducible nonnneg-ative matrices, Linear Algebra Appl., 361: 257–277, 2003.[24] S. SAVCHENKO, Typical changes in spectral properties under perturbations by a rank-one operator,Mat. Zametki, 74: 590–602, 2003., (Russian). Translation in Mathematical Notes, 74: 557–568, 2003.[25] S. SAVCHENKO, On the change in the spectral properties of a matrix under a perturbation of a suf-ficiently low rank, Funktsional. Anal. i Prilozhen, 38: 85–88, 2004., (Russian). Translation in Funct.Anal. Appl. 38: 69–71, 2004.[26] R. THOMPSON, The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil,Linear Algebra Appl., 14: 135–177, 1976.Operators and Matriceswww.ele-math.comoam@ele-math.com

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