The polar decomposition of block companion matrices

Let L(λ) = Inλm + Am−1λm−1 + …+A1λ + A0 be an n × n monic matrix polynomial, and let CL be the corresponding block companion matrix. In this note, we extend a known result on scalar polynomials to obtain a formula for the polar decomposition of CL when the matrices A0 and Σj=1m−1AjA*j are nonsingular.