Zero points of quadratic matrix polynomials

Opfer, Gerhard, and Janovská, Drahoslava. "Zero points of quadratic matrix polynomials." Applications of Mathematics 2013. Prague: Institute of Mathematics AS CR, 2013. 168-176. <http://eudml.org/doc/287800>.

@inProceedings{Opfer2013,
abstract = {Our aim is to classify and compute zeros of the quadratic two sided matrix polynomials, i.e. quadratic polynomials whose matrix coefficients are located at both sides of the powers of the matrix variable. We suppose that there are no multiple terms of the same degree in the polynomial $\mathbf \{p\}$, i.e., the terms have the form $\{\mathbf \{A\}\}_j\{\mathbf \{X\}\}^j\{\mathbf \{B\}\}_j$, where all quantities $\{\mathbf \{X\}\},\{\mathbf \{A\}\}_j,\{\mathbf \{B\}\}_j,j=0,1,\ldots ,N,$ are square matrices of the same size. Both for classification and computation, the essential tool is the description of the polynomial $\mathbf \{p\}$ by a matrix equation $\mathbf \{P\}(\mathbf \{X\}) := \mathbf \{A\}(\mathbf \{X\})\mathbf \{X\}+\mathbf \{B\}(\mathbf \{X\})$, where $\mathbf \{A\}(\mathbf \{X\})$ is determined by the coefficients of the given polynomial $\mathbf \{p\}$ and $\mathbf \{P\}, \mathbf \{X\},\mathbf \{B\}$ are real column vectors. This representation allows us to classify five types of zero points of the polynomial $\mathbf \{p\}$ in dependence on the rank of the matrix $\mathbf \{A\}$. This information can be for example used for finding all zeros in the same class of equivalence if only one zero in that class is known. For computation of zeros, we apply Newtons method to $\mathbf \{P\}(\mathbf \{X\}) = \mathbf \{0\}.$},
author = {Opfer, Gerhard, Janovská, Drahoslava},
booktitle = {Applications of Mathematics 2013},
keywords = {Cayley-Hamilton theorem; quadratic matrix polynomial; Newton’s method; matrix equation; zero points},
location = {Prague},
pages = {168-176},
publisher = {Institute of Mathematics AS CR},
title = {Zero points of quadratic matrix polynomials},
url = {http://eudml.org/doc/287800},
year = {2013},
}

TY - CLSWK
AU - Opfer, Gerhard
AU - Janovská, Drahoslava
TI - Zero points of quadratic matrix polynomials
T2 - Applications of Mathematics 2013
PY - 2013
CY - Prague
PB - Institute of Mathematics AS CR
SP - 168
EP - 176
AB - Our aim is to classify and compute zeros of the quadratic two sided matrix polynomials, i.e. quadratic polynomials whose matrix coefficients are located at both sides of the powers of the matrix variable. We suppose that there are no multiple terms of the same degree in the polynomial $\mathbf {p}$, i.e., the terms have the form ${\mathbf {A}}_j{\mathbf {X}}^j{\mathbf {B}}_j$, where all quantities ${\mathbf {X}},{\mathbf {A}}_j,{\mathbf {B}}_j,j=0,1,\ldots ,N,$ are square matrices of the same size. Both for classification and computation, the essential tool is the description of the polynomial $\mathbf {p}$ by a matrix equation $\mathbf {P}(\mathbf {X}) := \mathbf {A}(\mathbf {X})\mathbf {X}+\mathbf {B}(\mathbf {X})$, where $\mathbf {A}(\mathbf {X})$ is determined by the coefficients of the given polynomial $\mathbf {p}$ and $\mathbf {P}, \mathbf {X},\mathbf {B}$ are real column vectors. This representation allows us to classify five types of zero points of the polynomial $\mathbf {p}$ in dependence on the rank of the matrix $\mathbf {A}$. This information can be for example used for finding all zeros in the same class of equivalence if only one zero in that class is known. For computation of zeros, we apply Newtons method to $\mathbf {P}(\mathbf {X}) = \mathbf {0}.$
KW - Cayley-Hamilton theorem; quadratic matrix polynomial; Newton’s method; matrix equation; zero points
UR - http://eudml.org/doc/287800
ER -