A permuted factors approach for the linearization of polynomial matrices

In [1] and [2] a new family of companion forms associated to a regular polynomial matrix T (s) has been presented, using products of permutations of n elementary matrices, generalizing similar results presented in [3] where the scalar case was considered. In this paper, extending this “permuted factors” approach, we present a broader family of companion like linearizations, using products of up to n(n− 1)/2 elementary matrices, where n is the degree of the polynomial matrix. Under given conditions, the proposed linearizations can be shown to consist of block elements where the coefficients of the polynomial matrix appear intact. Additionaly we provide a criterion for those linearizations to be block symmetric. We also illustrate several new block symmetric linearizations strictly equivalent to the original polynomial matrix T (s) where in some of them, the constraint of nonsingularity of the constant term and the coefficient of maximum degree is not a prerequisite.