In this paper, the notion of rank-k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for ϵ > 0; the notion of Birkhoff-James approximate orthogonality sets for ϵ-higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed denitions yield a natural genera...

In this paper, the notion of rank−k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for > 0, the notion of Birkhoff-James approximate orthogonality sets for −higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed definitions yield a natural general...

Classical orthogonal polynomials and matrix polynomials being orthogonal with respect to some Hermitean positive deenite matrix of measures share several properties , e.g., three term recurrencies, Christooel{Darboux formulas; there are connections to the triangular decomposition of the (inverse) moment matrix and to eigenvalue{problems for the banded matrix of recurrence coeecients. Also, a co...

Matrix methods for the computation of bounds range a complex polynomial and its modulus over rectangular region in plane are presented. The approach relies on expansion given into Bernstein polynomials. results extended to multivariate polynomials rational functions.

We present an algorithm for the computation of a shifted Popov Normal Form of a rectangular polynomial matrix. For specific input shifts, we obtain methods for computing the matrix greatest common divisor of two matrix polynomials (in normal form) or such polynomial normal form computation as the classical Popov form and the Hermite Normal Form. The method is done by embedding the problem of co...

We present a Las Vegas probabalistic algorithm for reducing the computation of Hermite normal forms of rectangular polynomial matrices. In particular, the problem of computing the Hermite normal form of a rectangular m n matrix (with m > n) reduces to that of computing the Hermite normal form of a matrix of size (n + 1) n having entries of similar coeecient size and degree. The main cost of the...

in this paper, we give some determinantal and permanental representations of generalized lucas polynomials, which are a general form of generalized bivariate lucas p-polynomials, ordinary lucas and perrin sequences etc., by using various hessenberg matrices. in addition, we show that determinant and permanent of these hessenberg matrices can be obtained by using combinations. then we show, the ...

We construct the ($\beta$-deformed) partition function hierarchies with $W$-representations. Based on $W$-representations, we analyze superintegrability property and derive their character expansions respect to Schur functions Jack polynomials, respectively. Some well known superintegrable matrix models such as Gaussian hermitian one-matrix model (in external field), $N\times N$ complex model, ...