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 \(A - μ C\). We first present fundamental properties of the 2DEVP, such as the existence and variational characterizations of 2D-eigenvalues, 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.