Linearizations, realization, and scalar products for regular matrix polynomials (Q1379105)

The authors are concerned with polynomials \(L(\lambda)= \sum^\ell_{i=0} \lambda^i A_i\), where the \(A_i\) are \(n\times n\) complex matrices and \(\text{det} L (\lambda) \neq 0\). The linearizations of \(L(\lambda)\) are certain \(\ell n\times \ell n\) pencils \(\lambda G-A\) that reflect its properties. In order to reduce considerations to the case where the leading matrix coefficient \(A_\ell\) is nonsingular, the authors make a systematic study of the linearizations of the shifted polynomials \(L_\alpha (\lambda): =L(\alpha- \lambda)\) with \(L(\alpha)\neq 0\). One byproduct is a new realization of \(L(\lambda)^{-1}\) in terms of a matrix pencil. The self-adjoint case where \(A^*_i= A_i\), \(i=0, \dots, \ell\), is also studied. Here the aim is to determine explicitly Hermitian matrices with respect to which given linearizations are self-adjoint.