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Sharp lower bounds for the dimension of linearizations of matrix polynomials. (English) Zbl 1173.15014

A standard way of dealing with matrix polynomial eigenvalue problems is to use linearizations. In a recent work, R. Byers, V. Mehrmann and H. Xu [Linear Algebra Appl. 429, No. 10, 2373–2400 (2008; Zbl 1155.65026)] defined and studied linearizations of dimensions smaller than the classical ones. In the present paper the authors provide lower bounds for the dimensions of linearizations and string linearizations of a given \(m\times n\) matrix polynomial. Also, particular linearizations are constructed for which these bounds are attained. It is also shown that strong linearization of \(n\times n\) regular matrix polynomial of degree \(\ell\) must have dimension \(n\ell \times n\ell\).

MSC:

15A54 Matrices over function rings in one or more variables
15A18 Eigenvalues, singular values, and eigenvectors
15A21 Canonical forms, reductions, classification
15A22 Matrix pencils
65F15 Numerical computation of eigenvalues and eigenvectors of matrices

Citations:

Zbl 1155.65026
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