Inertia, Numerical Range, and Zeros of Quadratic Forms for Matrix Pencils

Abstract

Definite, semidefinite, and indefinite Hermitian and symmetric matrix pencils $P( A,B )$ are classified by their $l_{\bf C} $ and $l_{\bf R} $ numbers where $l_{\bf F} = \dim \operatorname{span} \{ X \in {\bf F}^n :x^ * Ax = x^ * Bx = 0 \}$. Using ideas from numerical range theory, it is proved for ${\bf F} = {\bf C}$ that $P( A,B )$ is a definite pencil if and only if $l_{\bf C} = 0,P( A,B )$ is an indefinite pencil if and only if $l_{\bf C} = n$, while $P( A,B )$ is a semidefinite pencil if and only if $0 < l_{\bf C} < n$. In contrast, for ${\bf F} = {\bf R}$ the $l_{\bf R} $ number for indefinite pencils can be as low as $n - 2$. In the cases for ${\bf F} = {\bf R}$ with $l_{\bf R} = n - 2$ or $n - 1$, the Kronecker canonical form theory is used to describe sets of generators for indefinite and semidefinite pencils $P( A,B )$.