This paper formulates the problem of determining the degrees of all polynomial entries in a minimal polynomial basis of a polynomial matrix by using only the degree structure of the given matrix. Using a block Toeplitz matrix structure corresponding to the given polynomial matrix, it is shown that the degrees of the elements in its minimal polynomial basis, in the generic case, depend only on t...

The Gould–Hopper–Laguerre–Sheffer matrix polynomials were initially studied using operational methods, but in this paper, we investigate them techniques. By leveraging properties of Pascal functionals and Wronskian matrices, derive several identities for these polynomials, including recurrence relations. It is highlighted that identities, acquired via techniques, are distinct from the ones obta...

We study the problem of dictionary learning for signals that can be represented as polynomials or polynomial matrices, such as convolutive signals with time delays or acoustic impulse responses. Recently, we developed a method for polynomial dictionary learning based on the fact that a polynomial matrix can be expressed as a polynomial with matrix coefficients, where the coefficient of the poly...

8 Matrices 16 8.1 Representation of vector spaces and linear transformations . . . . . . . . . . . . . . . 16 8.2 Similarity transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 8.3 Direct sums of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 8.4 The companion matrix of a polynomial . . . . . . . . . . . . . . . . . . . . . . . ....