Typical matrix eigenvalue problems, quadratic or linear, are best formulated as pencils \((A,M)\) in which both \(A\) and \(M\) are real and symmetric. This fact is emphasized in the paper through a set of physical examples. Then, the canonical forms are used to explain the role of the sign characteristic attached to real eigenvalues. The Rayleigh quotient is analyzed to describe real eigenvalues and it seems plausible that the class of definite Hermitian pencils is the broadest extension of the class of Hermitian matrices that retains the classical properties. 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 also mentioned.