some properties of the block toeplitz-hessenberg matrices

In this paper, we have a t × t matrix polynomial that is defined byl(λ)= λ^r -∑(r,j=1)λ ^r-j cj, cj ∈ c ^t*t, j=1,...,rof 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.