Abstract
One investigates the spectral problem for polynomial λ-matrices of the general form (regular and singular). One establishes a relation between the elements of the complete spectral structure of the λ-matrix (i.e., its elementary divisors, its Jordan chains of vectors, its minimal indices and polynomial solutions) and of the matrices which occur in its transformation to the Smith canonical form. One establishes a correspondence between the complete spectral structures of a λ-matrix and of three linear accompanying matrix pencils. One notes the possibility of reducing the solution of the spectral problem for a λ-matrix to the solution of a similar problem for its accompanying pencil. One gives a factorization of a λ-matrix which allows to represent it by any of its considered accompanying pencils or by its Kronecker canonical form.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
F. R. Gantmakher (Gantmacher), The Theory of Matrices, Chelsea, New York (1959).
M. Langer, “Über Lancaster's Zerlegung von Matrizen-Scaren,” Arch. Rat. Mech. Anal.,29, No. 1, 75–80 (1968).
P. Lancaster and H. K. Wimmer, “Zur Theorie der λ -Matrizen,” Math. Nachr.,68, 325–330 (1975).
G. D. Forney, Jr., “Minimal bases of rational vector spaces with applications to multi-variable linear systems,” SIAM J. Control Optim.,13, No. 3, 493–520 (1975).
I. Ts. Gokhberg and E. I. Sigal, “An operator generalization of the logarithmic residue theorem and the theorem of Rouché,” Mat. Sb.,84, (126), No. 4, 607–629 (1971).
G. Peters and J. H. Wilkinson, “Ax=λBx and the generalized eigenproblem,” SIAM J. Numer. Anal.,7, No. 4, 479–492 (1970).
J. E. Dennis, J. F. Traub, and R. P. Weber, “On the matrix polynomials, λ-matrices and block eigenvalue problems,” Cornell University, Carnegie-Mellon University, Computer Science Dept. Technical Report (1971).
A. I. Belinskii, “Some methods of investigation of generalized eigenvalue problems,” Candidate's Dissertation, Lvov (1972).
I. Gohberg, P. Lancaster, and L. Rodman, “Spectral analysis of matrix polynomials, I. Canonical forms and divisors,” Linear Algebra Appl.,20, No. 1, 1–44 (1978).
P. Van Dooren, “The generalized eigenstructure problem. Application in linear systems theory,“ Thesis. Katholieke Universiteit te Leuven (1979).
P. Lancaster, Lambda-Matrices and Vibrating Systems, Pergamon Press, Oxford (1966),
H. K. Wimmer, “A Jordan factorization theorem for polynomial matrices,” Proc. Am. Math. Soc.,75, No. 2, 201–206 (1979).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, pp. 180–194, 1981.
Rights and permissions
About this article
Cite this article
Khazanov, V.B. Spectral properties of λ-matrices. J Math Sci 24, 121–132 (1984). https://doi.org/10.1007/BF01230274
Issue Date:
DOI: https://doi.org/10.1007/BF01230274