Spectral decomposition of real symmetric quadratic 𝜆-matrices and its applications

Chu, Moody; Xu, Shu-Fang

doi:10.1090/S0025-5718-08-02128-5

Spectral decomposition of real symmetric quadratic $\lambda$-matrices and its applications
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by Moody T. Chu and Shu-Fang Xu PDF

Math. Comp. 78 (2009), 293-313 Request permission

Abstract:

Spectral decomposition provides a canonical representation of an operator over a vector space in terms of its eigenvalues and eigenfunctions. The canonical form often facilitates discussions which, otherwise, would be complicated and involved. Spectral decomposition is of fundamental importance in many applications. The well-known GLR theory generalizes the classical result of eigendecomposition to matrix polynomials of higher degrees, but its development is based on complex numbers. This paper modifies the GLR theory for the special application to real symmetric quadratic matrix polynomials, $\mathcal {Q}(\lambda )=M \lambda ^{2} + C \lambda + K$, $M$ nonsingular, subject to the specific restriction that all matrices in the representation be real-valued. It is shown that the existence of the real spectral decomposition can be characterized through the notion of real standard pair which, in turn, can be constructed from the spectral data. Applications to a variety of challenging inverse problems are discussed.

References

Angelika Bunse-Gerstner, Ralph Byers, and Volker Mehrmann, A chart of numerical methods for structured eigenvalue problems, SIAM J. Matrix Anal. Appl. 13 (1992), no. 2, 419–453. MR 1152761, DOI 10.1137/0613028
Joao B. Carvalho, Biswa N. Datta, Wen-Wei Lin, and Chern-Shuh Wang, Symmetry preserving eigenvalue embedding in finite-element model updating of vibrating structures, J. Sound Vibration 290 (2006), no. 3-5, 839–864. MR 2196623, DOI 10.1016/j.jsv.2005.04.030
M. T. Chu, B. Datta, W. W. Lin and S. F. Xu, The spill-over phenomenon in quadratic model updating, AIAA J., 46(2008), 420–428.
M. T. Chu and N. Del Buono, Total decoupling of a general quadratic pencil, Pt. I: Theory, J. Sound Vibration, 309(2008), 96-111.
M. T. Chu and N. Del Buono, Total decoupling of a general quadratic pencil, Pt. II: Structure preserving isospectral flows, J. Sound Vibration, 309(2008), 112–128.
Moody T. Chu and Gene H. Golub, Inverse eigenvalue problems: theory, algorithms, and applications, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2005. MR 2263317, DOI 10.1093/acprof:oso/9780198566649.001.0001
S. D. Garvey, M. I. Friswell, and U. Prells, Co-ordinate transformations for second order systems. I. General transformations, J. Sound Vibration 258 (2002), no. 5, 885–909. MR 1944645, DOI 10.1006/jsvi.2002.5165
S. D. Garvey, M. I. Friswell, and U. Prells, Co-ordinate transformations for second order systems. II. Elementary structure-preserving transformations, J. Sound Vibration 258 (2002), no. 5, 911–930. MR 1944646, DOI 10.1006/jsvi.2002.5166
F. R. Gantmacher, The theory of matrices. Vol. 1, AMS Chelsea Publishing, Providence, RI, 1998. Translated from the Russian by K. A. Hirsch; Reprint of the 1959 translation. MR 1657129
I. Gohberg, P. Lancaster, and L. Rodman, Spectral analysis of selfadjoint matrix polynomials, Ann. of Math. (2) 112 (1980), no. 1, 33–71. MR 584074, DOI 10.2307/1971320
I. Gohberg, P. Lancaster, and L. Rodman, Matrix polynomials, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. MR 662418
I. Gohberg, P. Lancaster, and L. Rodman, Matrices and indefinite scalar products, Operator Theory: Advances and Applications, vol. 8, Birkhäuser Verlag, Basel, 1983. MR 859708
P. R. Halmos, What does the spectral theorem say?, Amer. Math. Monthly 70 (1963), 241–247. MR 150600, DOI 10.2307/2313117
Nicholas J. Higham, D. Steven Mackey, Niloufer Mackey, and Françoise Tisseur, Symmetric linearizations for matrix polynomials, SIAM J. Matrix Anal. Appl. 29 (2006/07), no. 1, 143–159. MR 2288018, DOI 10.1137/050646202
Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1990. Corrected reprint of the 1985 original. MR 1084815
Yuen-Cheng Kuo, Wen-Wei Lin, and Shu-Fang Xu, Solutions of the partially described inverse quadratic eigenvalue problem, SIAM J. Matrix Anal. Appl. 29 (2006/07), no. 1, 33–53. MR 2288012, DOI 10.1137/05064134X
P. Lancaster, Expressions for damping matrices in linear vibration problems, J. Aero-Space Sci., 28(1961), pp. 256.
Peter Lancaster, Lambda-matrices and vibrating systems, International Series of Monographs in Pure and Applied Mathematics, Vol. 94, Pergamon Press, Oxford-New York-Paris, 1966. MR 0210345
Peter Lancaster, Isospectral vibrating systems. I. The spectral method, Linear Algebra Appl. 409 (2005), 51–69. MR 2169546, DOI 10.1016/j.laa.2005.01.029
Peter Lancaster, Inverse spectral problems for semisimple damped vibrating systems, SIAM J. Matrix Anal. Appl. 29 (2006/07), no. 1, 279–301. MR 2288026, DOI 10.1137/050640187
P. Lancaster and J. Maroulas, Inverse eigenvalue problems for damped vibrating systems, J. Math. Anal. Appl. 123 (1987), no. 1, 238–261. MR 881543, DOI 10.1016/0022-247X(87)90306-4
P. Lancaster and U. Prells, Inverse problems for damped vibrating systems, J. Sound Vibration 283 (2005), no. 3-5, 891–914. MR 2136391, DOI 10.1016/j.jsv.2004.05.003
Peter Lancaster and Leiba Rodman, Canonical forms for Hermitian matrix pairs under strict equivalence and congruence, SIAM Rev. 47 (2005), no. 3, 407–443. MR 2178635, DOI 10.1137/S003614450444556X
P. Lancaster and L. Rodman, Canonical forms for symmetric/skew-symmetric real matrix pairs under strict equivalence and congruence, Linear Algebra Appl. 406 (2005), 1–76. MR 2156428, DOI 10.1016/j.laa.2005.03.035
Peter Lancaster and Qiang Ye, Inverse spectral problems for linear and quadratic matrix pencils, Proceedings of the Victoria Conference on Combinatorial Matrix Analysis (Victoria, BC, 1987), 1988, pp. 293–309. MR 960152, DOI 10.1016/0024-3795(88)90252-2
Chris Paige and Charles Van Loan, A Schur decomposition for Hamiltonian matrices, Linear Algebra Appl. 41 (1981), 11–32. MR 649714, DOI 10.1016/0024-3795(81)90086-0
Uwe Prells and Peter Lancaster, Isospectral vibrating systems. II. Structure preserving transformations, Operator theory and indefinite inner product spaces, Oper. Theory Adv. Appl., vol. 163, Birkhäuser, Basel, 2006, pp. 275–298. MR 2215867, DOI 10.1007/3-7643-7516-7_{1}2
Françoise Tisseur and Karl Meerbergen, The quadratic eigenvalue problem, SIAM Rev. 43 (2001), no. 2, 235–286. MR 1861082, DOI 10.1137/S0036144500381988