Zero points of quadratic matrix polynomials. (English) Zbl 1340.65080

Opfer, Gerhard; Janovská, Drahoslava

Brandts, J. (ed.) et al., Proceedings of the international conference ‘Applications of mathematics’, Prague, Czech Republic, May 15–17, 2013. In honor of the 70th birthday of Karel Segeth. Prague: Academy of Sciences of the Czech Republic, Institute of Mathematics (ISBN 978-80-85823-61-5). 168-176 (2013).

Summary: 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 \(\mathbf X, \mathbf A_j, \mathbf B_j, j=0,1,\dots ,N\), 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 \(\mathbf P(\mathbf X)=0\).
For the entire collection see [Zbl 1277.00032].

MSC:

65F30	Other matrix algorithms (MSC2010)
65H04	Numerical computation of roots of polynomial equations
15A54	Matrices over function rings in one or more variables

Keywords:

Cayley-Hamilton theorem; quadratic matrix polynomial; Newton’s method; matrix equation; zero points

Software:

Matlab