The author derives analogues of the classical min-max and max-min characterizations of the eigenvalues of an \(n\times n\) Hermitian matrix \(A\). In these analogues, the Rayleigh quotient is replaced by \((Au,v)/(u,v)\), with \((u,v)>0\), where \(u\), \(v\) belong to a certain subspace of \({\mathbb C}^n\) isomorphic to \({\mathbb R}^n\). Corresponding results for singular values of a general matrix are illustrated by a numerical example arising from the description of a simple damped vibrating system.