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Keywords:
Cayley-Hamilton theorem; quadratic matrix polynomial; Newton’s method; matrix equation; zero points
Cayley-Hamilton theorem; quadratic matrix polynomial; Newton’s method; matrix equation; zero points
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 ${\mathbf{X}},{\mathbf{A}}_j,{\mathbf{B}}_j,j=0,1,\ldots,N,$ 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 $\mathbf{P}(\mathbf{X}) = \mathbf{0}.$
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