Linearization of regular matrix polynomials
نویسنده
چکیده
This note contains a short review of the notion of linearization of regular matrix polynomials. The objective is clarification of this notion when the polynomial has an “eigenvalue at infinity”. The theory is extended to admit reduction by locally unimodular analytic matrix functions.
منابع مشابه
Higher numerical ranges of matrix polynomials
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...
متن کاملSome Results on the Field of Values of Matrix Polynomials
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 ...
متن کاملA Note on Weak and Strong Linearizations of Regular Matrix Polynomials
A paper of Tan and Pugh [TP] raises the question of ambiguity in a frequently used form of linearization when applied to regular matrix polynomials. Here, further insight into this question is provided, as well as a reminder of a stronger form of linearization (for which ambiguities are removed) introduced by Gohberg et al. [GKL]. Let A0, A1, . . . , An ∈ C , and define the matrix polynomial L(...
متن کاملEla Linearizations of Singular Matrix Polynomials and the Recovery of Minimal Indices∗
A standard way of dealing with a regular matrix polynomial P (λ) is to convert it into an equivalent matrix pencil – a process known as linearization. Two vector spaces of pencils L1(P ) and L2(P ) that generalize the first and second companion forms have recently been introduced by Mackey, Mackey, Mehl and Mehrmann. Almost all of these pencils are linearizations for P (λ) when P is regular. Th...
متن کاملTriangularizing Quadratic Matrix Polynomials
We show that any regular quadratic matrix polynomial can be reduced to an upper triangular quadratic matrix polynomial over the complex numbers preserving the finite and infinite elementary divisors. We characterize the real quadratic matrix polynomials that are triangularizable over the real numbers and show that those that are not triangularizable are quasi-triangularizable with diagonal bloc...
متن کامل