VARIATIONAL CHARACTERIZATION AND RAYLEIGH QUOTIENT ITERATION OF 2D EIGENVALUE PROBLEM WITH APPLICATIONS

A two dimensional eigenvalue problem (2DEVP) of a Hermitian matrix pair (A, C) is introduced in this paper. The 2DEVP can be regarded as a linear algebra formulation of the well-known eigenvalue optimization problem of the parameter matrix A - \mu C. We first present fundamental properties of the 2DEVP, such as the existence and variational characterizations of 2Deigenvalues, and then devise a Rayleigh quotient iteration (RQI)-like algorithm, 2DRQI in short, for computing a 2D-eigentriplet of the 2DEVP. The efficacy of the 2DRQI is demonstrated by large scale eigenvalue optimization problems arising from the minmax of Rayleigh quotients and the distance to instability of a stable matrix.