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An inversion formula for a matrix polynomial about a (unit) root. (English) Zbl 1219.15005

The authors provide a solution to the problem of a closed-form representation for the inverse of a matrix polynomial about a unit root. This is accomplished by resorting to a Laurent expansion in matrix notation, whose principal-part coefficients turn out to depend on the non-null derivatives of the adjoint and the determinant of the matrix polynomial at the root. They also derive certain basic relationships between the principal-part structure and the rank properties of algebraic function of the matrix polynomial at the unit root as well as informative closed-form expressions for the leading coefficient matrices of the matrix-polynomial inverse.

MSC:

15A09 Theory of matrix inversion and generalized inverses
15A54 Matrices over function rings in one or more variables
15A15 Determinants, permanents, traces, other special matrix functions
15A18 Eigenvalues, singular values, and eigenvectors
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
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[1] DOI: 10.1137/S0895479898337555 · Zbl 0985.15005 · doi:10.1137/S0895479898337555
[2] DOI: 10.1093/0198288107.001.0001 · Zbl 0937.62650 · doi:10.1093/0198288107.001.0001
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