Linearization of matrix polynomials expressed in polynomial bases

Abstract

Companion matrices of matrix polynomials L(λ) (with possibly singular leading coefficient) are a familiar tool in matrix theory and numerical practice leading to so-called "linearizations" λB − A of the polynomials. Matrix polynomials as approximations to more general matrix functions lead to the study of matrix polynomials represented in a variety of classical systems of polynomials, including orthogonal systems and Lagrange polynomials, for example. For several such representations, it is shown how to construct (strong) linearizations via analogous companion matrix pencils. In case L(λ) has Hermitian or alternatively complex symmetric coefficients, the determination of linearizations λB − A with A and B Hermitian or complex symmetric is also discussed.

Möbius transformations of matrix polynomials

“…For example, suppose P is a matrix polynomial whose Jordan characteristic at λ 0 is J (P , λ 0 ) = (0, 0, 0, 0, 1, 3, 4). Then the truncated Jordan characteristic of P at λ 0 will be just (1,3,4). We define this formally as follows.…”

“…For example, the conditions in Definition 4.9 can be relaxed as in [40] to allow E λ 0 (λ) and F λ 0 (λ) to be any matrix functions that are invertible at λ 0 and analytic in a neighborhood of λ 0 (e.g., rational matrices whose determinants have no zeroes or poles at λ 0 ). This extension was used in [3] to study linearizations.…”

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

“…This could be viewed as another kind of structure preservation, i.e., a preservation of the polynomial basis. Although there are precedents for doing this for scalar polynomials in [10], and even earlier in [33], the first serious effort in this direction for matrix polynomials was [5] and the earlier [17], where concrete templates for producing strong linearizations were provided, one for each of several classical polynomial bases, including Chebyshev, Newton, Lagrange, and Bernstein bases. This work has been used in [26], as part of a Chebyshev interpolation method for solving non-polynomial nonlinear eigenproblems.…”

Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis

“…The linearizations discussed in [2] for general polynomial bases satisfying a threeterm recurrence relation all admit a highly structured block LU factorization similar to the one discussed above. It is therefore likely that the TOAR method can be extended to this more general setting.…”