J-PSEUDO-SPECTRAL AND J-INNER-PSEUDO-OUTER FACTORIZATIONS FOR MATRIX POLYNOMIALS

For a comonic polynomial L(lambda) and a selfadjoint invertible matrix J the following two factorization problems are considered: firstly, w e parametrize all comonic polynomials R(lambda) such that L(<(lambda)o ver bar>)JL(lambda) = R(<(lambda)over bar>)*JR(lambda). Secondly, if it exists, we give the J-inner-pseudo-outer factorization L(lambda) = Theta(lambda)R(lambda), where Theta(lambda) is J-inner and R(lambda) i s a comonic pseudo-outer polynomial. We shall also consider these prob lems with additional restrictions on the pole structure and/or zero st ructure of R(lambda). The analysis of these problems is based on the s olution of a general inverse spectral problem for rational matrix func tions, which consists of finding the set of rational matrix functions for which two given pairs are extensions of their pole and zero pair, respectively.