Some properties of the block Toeplitz-Hessenberg matrices

In this paper, we have a t × t matrix polynomial that is defined by L(λ)= λ^r -∑(r,j=1)λ ^r-j Cj, Cj ∈ C ^t*t, j=1,...,r of degree r that σ(L) = {λ ∈ C 1 : det L(λ) = 0}. Here, we try to show a process for finding determinant of the block Toeplitz-Hessenberg matrices from matrix polynomials by the block companion matrices. Also, we generalize a new version of Trudi’s formula for the block Toeplitz-Hessenberg matrices. These tools are used to find eigenvalues, singularities, and stability and instability of systems.