Let $P(lambda)$ be an $n$-square complex matrix polynomial, and $1 leq k leq n$ be a positive integer. In this paper, some algebraic and geometrical properties of the $k$-numerical range of $P(lambda)$ are investigated. In particular, the relationship between the $k$-numerical range of $P(lambda)$ and the $k$-numerical range of its companion linearization is stated. Moreover, the $k$-numerical...

let $p(lambda)$ be an $n$-square complex matrix polynomial, and $1 leq k leq n$ be a positive integer. in this paper, some algebraic and geometrical properties of the $k$-numerical range of $p(lambda)$ are investigated. in particular, the relationship between the $k$-numerical range of $p(lambda)$ and the $k$-numerical range of its companion linearization is stated. moreover, the $k$-numerical ...

An extended innnite companion matrix ~ C 1 (D) and an innnite companion matrix C 1 (D) for a (nonmonic in general) matrix polynomial D is introduced and the nite companion matrix C(D) is generalized to the nonmonic case. These matrices generalize all properties of the innnite and nite companion (Frobenius) matrix corresponding to a scalar polynomial. In particular, C 1 (D) is a controllability ...

In this paper, the notions of pseudofield of values and joint pseudofield of values of matrix polynomials are introduced and some of their algebraic and geometrical properties are studied.  Moreover, the relationship between the pseudofield of values of a matrix polynomial and the pseudofield of values of its companion linearization is stated, and then some properties of the augmented field of ...

An explicit formula for the polar decomposition of an n n nonsingular companion matrix is derived. The proof involves the largest and smallest singular values of the companion matrix.

A method is developed to compute the zeros of a quaternion polynomial with all terms of the form ok Xk. This method is based essentially in Niven’s algorithm (11, which consists of dividing the polynomial by a characteristic polynomial associated to a zero. The information about the trace and the norm of the zero is obtained by an original idea which requires the companion matrix associated to ...