Common eigenvalues, divisors, and multiples of matrix polynomials: A review. (English) Zbl 0627.15004
This paper gives an overview of some results obtained in the last decade using methods of spectral theory and concerning common multiples, common divisors, and common eigenvalues for matrix polynomials. This includes discussion of greatest common divisors and least common multiples of matrix polynomials in terms of spectral data and of resultant and Bézoutian matrices for matrix polynomials with descriptions of their kernels in terms of spectral data, as well as a guide to the literature.
Reviewer: J.A.Ball
MSC:
| 15A30 | Algebraic systems of matrices |
| 15A21 | Canonical forms, reductions, classification |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15-02 | Research exposition (monographs, survey articles) pertaining to linear algebra |
Keywords:
spectral theory; common multiples; common divisors; common eigenvalues; matrix polynomials; greatest common divisors; least common multiples; resultant; Bézoutian matricesReferences:
| [1] | Anderson, B. D.O.; Jury, E. I., Generalized Bezoutian and Sylvester matrices in multivariable linear control, IEEE Trans. Automat. Control, AC-21, 551-556 (1976) · Zbl 0332.93032 |
| [2] | Barnett, S., A note on the bezoutian matrix, SIAM J. Appl. Math., 22, 84-86 (1972) · Zbl 0245.15011 |
| [3] | Barnett, S.; Lancaster, P., Some properties of the Bezoutian for polynomial matrices, Linear and Multilinear Algebra, 9, 99-110 (1980) · Zbl 0504.15002 |
| [4] | Bitmead, R. R.; Kung, S. Y.; Anderson, B. D.O.; Kailath, T., Greatest common divisors via generalized Sylvester and Bezout matrices, IEEE Trans. Automat. Control, AC-23, 1043-1047 (1978) · Zbl 0389.93008 |
| [5] | Clancey, K. F.; Kon, B. A., The Bezoutian and the algebraic Riccati equation, Linear and Multilinear Algebra, 15, 265-278 (1984) · Zbl 0571.15007 |
| [6] | Cohen, N., On spectral analysis of matrix polynomials, (M.Sc. Thesis (1979), Weizmann Inst. of Science: Weizmann Inst. of Science Israel) |
| [7] | Elsner, L.; Lancaster, P., The spectral variation of pencils of matrices, J. Comput. Math., 3, 262-274 (1985) · Zbl 0593.15013 |
| [8] | Gantmacher, F. R., The Theory of Matrices (1959), Chelsea: Chelsea New York · Zbl 0085.01001 |
| [9] | Gohberg, I.; Heinig, G., The resultant matrix and its generalizations, I, The resultant operator for matrix polynomials (in Russian), Acta Sci. Math. (Szeged), 37, 41-61 (1975) · Zbl 0283.15009 |
| [10] | Gohberg, I.; Kaashoek, M. A.; Lerer; Rodman, L., Common multiples and common divisors of matrix polynomials, I, Spectral method, Indiana Univ. Math. J., 30, 321-356 (1981) · Zbl 0449.15015 |
| [11] | Gohberg, I.; Kaashoek, M. A.; Lerer, L.; Rodman, L., Common multiples and common divisors of matrix polynomials, II, Vandermonde and resultant matrices, Linear and Multilinear Algebra, 12, 159-203 (1982) · Zbl 0496.15014 |
| [12] | Gohberg, I.; Kaashoek, M. A.; Rodman, L., Spectral analysis of families of operator polynomials and a generalized Vandermonde matrix, I. The finite dimensional case, Adv. in Math., 3, 91-128 (1978) · Zbl 0453.15005 |
| [13] | Gohberg, I.; Kaashoek, M. A.; Rodman, L., Spectral analysis of families of operator polynomials and a generalized Vandermonde matrix, II. The infinite dimensional case, J. Funct. Anal., 30, 358-389 (1978) · Zbl 0404.47011 |
| [14] | Gohberg, I.; Lancaster, P.; Rodman, L., Matrix Polynomials (1982), Academic: Academic New York · Zbl 0482.15001 |
| [15] | Gohberg, I.; Lerer, L., Resultants of matrix polynomials, Bull. Amer. Math. Soc., 82, 275-277 (1976) · Zbl 0343.15011 |
| [16] | Gohberg, I.; Rodman, L., On spectral analysis of nonmonic matrix and operator polynomials, I. Reduction to monic polynomials, Israel J. Math., 30, 133-151 (1978) · Zbl 0396.47009 |
| [17] | Heinig, G., Verallgemeinerte Resultantbegriffe bei beliebigen Matrix büscheln, I, II, (Wiss. Z. Tech. Hochsch., 20 (1978), Karl-Marx-Stadt), 701-703 · Zbl 0453.15007 |
| [18] | Heinig, G., Resultante und Spektralverteilungs-probleme für Operatorpolynome, Math. Nachr., 91, 23-43 (1979) · Zbl 0464.47003 |
| [19] | Heinig, G.; Rost, K., Algebraic Methods for Toeplitz-Like Matrices and Operators (1984), Birkhäuser: Birkhäuser Basel · Zbl 0549.15013 |
| [20] | Kaashoek, M. A.; van der Mee, C.; Rodman, L., Analytic operator functions with compact spectrum, I, Spectral nodes, linearization and equivalence, Integral Equations Operator Theory, 4, 504-547 (1981) · Zbl 0476.47013 |
| [21] | Kaashoek, M. A.; van der Mee, C.; Rodman, L., Analytic operator functions with compact spectrum, II, Spectral pairs and factorization, Integral Equations Operator Theory, 5, 791-827 (1982) · Zbl 0493.47015 |
| [22] | Kaashoek, M. A.; van der Mee, C.; Rodman, L., Analytic operator functions with compact spectrum, III, Hilbert space case: Inverse problem and applications, J. Operator Theory, 10, 219-250 (1983) · Zbl 0536.47016 |
| [23] | Krein, M. G.; Naimark, M. A., The method of symmetric and hermitian forms in the theory of the separation of the roots of algebraic equations, Linear and Multilinear Algebra, 10, 265-308 (1981), (transl. of 1936 original in Russian). · Zbl 0584.12018 |
| [24] | Lancaster, P.; Maroulas, J., The kernel of the Bezoutian for operator polynomials, Linear and Multilinear Algebra, 17, 181-201 (1985) · Zbl 0583.47020 |
| [25] | Lancaster, P.; Tismenetsky, M., The Theory of Matrices, Second Edition, with Applications (1985), Academic: Academic Toronto · Zbl 0558.15001 |
| [26] | L. Lerer, L. Rodman, and M. Tismenetsky, Bezoutian and Schur-Cohen problem for operator polynomials, J. Math. Anal. Appl.; L. Lerer, L. Rodman, and M. Tismenetsky, Bezoutian and Schur-Cohen problem for operator polynomials, J. Math. Anal. Appl. · Zbl 0567.47016 |
| [27] | Lerer, L.; Tismenetsky, M., The Bezoutian and the eigenvalue-separation problem for matrix polynomials, Integral Equations Operator Theory, 5, 386-445 (1982) · Zbl 0504.47020 |
| [28] | Lerer, L.; Tismenetsky, M., Bezoutian for several matrix polynomials and matrix equations, (Tech. Report 88.145 (1984), IBM Scientific Center: IBM Scientific Center Haifa) · Zbl 0635.15014 |
| [29] | Lerer, L.; Woerdeman, H. J., resultant operators and the Bezout equation for analytic matrix functions, (Report No. 299 (1985), Subfaculteit Wiskunde en Informatica, Vrije Univ: Subfaculteit Wiskunde en Informatica, Vrije Univ Amsterdam) · Zbl 0648.47018 |
| [30] | Rodman, L., On existence of common multiples of monic operator polynomials, Integral Equations Operator Theory, 1, 400-411 (1978) · Zbl 0396.47010 |
| [31] | Rosenbrock, H. H., State-Space and Multivariable Theory (1970), Thomas Nelson: Thomas Nelson London · Zbl 0246.93010 |
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