Quasi Commutativity of Regular Matrix Polynomials: Resultant and Bezoutian
To Israel Gohberg, an outstanding mathematician, an inspiring teacher and a wonderful friend, on the occasion of his 80th birthday. Abstract. In a recent paper of I. Gohberg and the authors necessary and sufficient conditions are obtained in order that for two regular matrix polynomials L and M the dimension of the null space of the associate square resultant matrix is equal to the sum of the multiplicities of the common zeros of L and M, infinity included. The conditions are stated in terms of quasi commutativity. In the case of commuting matrix polynomials, in particular, in the scalar case, these conditions are automatically fulfilled. The proofs in the above paper are heavily based on the spectral theory of matrix polynomials. In the present paper a new proof is given of the sufficiency part of the result mentioned above. Here we use the connections between the Bezout and resultant matrices and a general abstract scheme for determining the null space of the Bezoutian of matrix polynomials which is based on a state space analysis of Bezoutians.
KeywordsMatrix polynomials common spectral data quasi commutativity block resultant matrices of square size Bezoutian state space analysis
Mathematics Subject Classification (2000)Primary 47A56 15A18 secondary 47B35 47B99
Unable to display preview. Download preview PDF.
- I. Gohberg, M.A. Kaashoek, and L. Lerer, Quasi-commutativity of entire matrix functions and the continuous analogue of the resultant, in: Modern operator theory and applications. The Igor Borisovich Simonenko Anniversary Volume, OT 170, Birkhäuser Verlag, Basel, 2007, pp. 101–106.Google Scholar
- I. Gohberg, M.A. Kaashoek, and L. Lerer, The continuous analogue of the resultant and related convolution operators, in: The extended field of operator theory (M.A. Dritschel, ed.), OT 171, Birkhäuser Verlag, Basel, 2007, pp. 107–127.Google Scholar
- I. Gohberg, M.A. Kaashoek and L. Rodman, Spectral analysis of families of operator polynomials and a generalized Vandermonde matrix, I. The finite dimensional case, in: Topics in Functional Analysis. Advances in Mathematics, Supplementary Studies, vol. 3, Academic Press, London 1978; pp. 91–128.Google Scholar
- I. Gohberg and L. Lerer, Matrix generalizations of M.G. Krein theorems on orthogonal polynomials. OT 34 Birkhäuser Verlag, Basel, 1988, pp. 137–202.Google Scholar
- B.L. van der Waerden, Modern Algbra, I and II (English translation), Frederick Ungar Publ. Co., New York, 1949 and 1950.Google Scholar