On Bézoutians of nonsquare matrix polynomials and inversion of matrices with nonsquare blocks (Q919058)

If M(\(\lambda\)), L(\(\lambda\)) are two possibly nonsquare matrix polynomials such that \(M(\lambda)L(\lambda)=0\), then \(A(\lambda,\mu)=(\lambda -\mu)^{-1}M(\lambda)L(\mu)\) is a matrix polynomial. The Bézoutian of M, L is the block matrix whose i,j entry is the coefficient of \(\lambda^ i\mu^ j\) in the expansion of A. In this paper the Bézoutian is used to establish formulae for the inverse of square matrices partitioned into nonsquare blocks. In particular, a generalization of the Gohberg-Semencul formula for the inverse of a generalized Toeplitz matrix is proved.