SYMMETRICAL MATRIX PENCILS

A significant number of matrix eigenvalue problems, quadratic or linear, are best reformulated as pencils (A, M) in which both A and M are real and symmetric. Some examples are given and then the canonical forms are re-examined to explain the role of the sign characteristic attached to real eigenvalues. In addition we examine the limitations on the use of the Rayleigh quotient functional (x, Ax)/(x, Mx) in describing the eigenvalues. This sheds new light on the class of definite pencils and the stability of their eigenvalues under perturbations. The reduction of indefinite pencils to useful sparse forms is mentioned.