Determining the inertia of a matrix pencil as a function of the parameter

AbstractWe determine the inertia of a linear real symmetric matrix pencil A(t)=A−tB of order n as a function of the parameter t. Finding the critical values where the inertia of A(t) changes is reduced to determining the real eigenvalues of a (not necessarily symmetric) matrix. The order of this matrix is at most r+s−r1, where r, s, and r1 are the ranks of A, B, and [A B], respectively. We illustrate the method by means of a numerical example. Then we reduce determining the inertia of a quadratic real symmetric matrix pencil A(t)=A −tB1−t2B2 to the linear case. Our results are extensions of those by Caron and Gould