# Positive semidefinite interval of matrix pencil and its applications for the generalized trust region subproblems

## TL;DR

This paper characterizes the positive semidefinite interval of matrix pencils and applies these findings to simplify solving the hard case of generalized trust region subproblems.

## Contribution

It provides a detailed analysis of the positive semidefinite interval of matrix pencils, especially when matrices are not simultaneously diagonalizable, and applies this to improve GTRS solutions.

## Key findings

- The interval $I_{\succeq}(A,B)$ can be empty, a singleton, or an interval.
- When both matrices are singular, the interval can be decomposed into block diagonals.
- The approach simplifies solving the hard case of GTRS by reducing it to linear systems or smaller GTRS problems.

## Abstract

We are concerned with finding the set $I_{\succeq}(A,B)$ of real values $\mu$ such that the matrix pencil $A+\mu B$ is positive semidefinite. If $A, B$ are not simultaneously diagonalizable via congruence (SDC), $I_{\succeq}(A,B)$ either is empty or has only one value $\mu.$ When $A, B$ are SDC, $I_{\succeq}(A,B),$ if not empty, can be a singleton or an interval. Especially, if $I_{\succeq}(A,B)$ is an interval and at least one of the matrices is nonsingular then its interior is the positive definite interval $I_{\succ}(A,B).$ If $A, B$ are both singular, then even $I_{\succeq}(A,B)$ is an interval, its interior may not be $I_{\succ}(A,B),$ but $A, B$ are then decomposed to block diagonals of submatrices $A_1, B_1$ with $B_1$ nonsingular such that $I_{\succeq}(A,B)=I_{\succeq}(A_1,B_1).$ Applying $I_{\succeq}(A,B),$ the hard-case of the generalized trust-region subproblem (GTRS) can be dealt with by only solving a system of linear equations or reduced to the easy-case of a GTRS of smaller size.