From matrix polynomial to determinant of block Toeplitz–Hessenberg matrix

This paper concerns the study of matrix polynomials of arbitrary degree. In terms of L(λ )=λ ^rI-∑ _j=1^rλ ^r-j𝒞_j with or without commuting coefficients ( 𝒞_i𝒞_j=𝒞_j𝒞_i, or 𝒞_i𝒞_j𝒞_j𝒞_i for 𝒞_i∈ℂ^t× t, i,j=1,… , r ) by determinant of block Toeplitz-Hessenberg matrices.