2D Eigenvalue Problems I: Existence and Number of Solutions

11/19/2019

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by   Yangfeng Su, et al.

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A two dimensional eigenvalue problem (2DEVP) of a Hermitian matrix pair (A, C) is introduced in this paper. The 2DEVP can be viewed as a linear algebraic formulation of the well-known eigenvalue optimization problem of the parameter matrix H(μ) = A - μ C. We present fundamental properties of the 2DEVP such as the existence, the necessary and sufficient condition for the finite number of 2D-eigenvalues and variational characterizations. We use eigenvalue optimization problems from the quadratic constrained quadratic program and the computation of distance to instability to show their connections with the 2DEVP and new insights of these problems derived from the properties of the 2DEVP.