An inversion formula for a matrix polynomial about a (unit) root

Abstract

A solution to the problem of a closed-form representation for the inverse of a matrix polynomial about a unit root is provided 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. Some basic relationships between principal-part structure and 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 are established.

Keywords:

• matrix polynomial inversion
• Laurent expansion in matrix form
• adjoint and determinant derivatives

AMS Subject Classifications::

• 15A09
• 15A15
• 15A18
• 41A58
• 47A56