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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 $A, B$ arenot 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 thematrices is nonsingular then its interior is the positive definite interval$I_{\succ}(A,B).$ If $A, B$ are both singular, then even $I_{\succeq}(A,B)$ isan interval, its interior may not be $I_{\succ}(A,B),$ but $A, B$ are thendecomposed to block diagonals of submatrices $A_1, B_1$ with $B_1$ nonsingularsuch 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 dealtwith by only solving a system of linear equations or reduced to the easy-caseof a GTRS of smaller size.

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