Analytical solutions to some generalized matrix eigenvalue problems
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We present analytical solutions to two classes of generalized matrix eigenvalue problems (GMEVPs) which naturally cover the matrix eigenvalue problems (MEVPs) with matrices such as the tridiagonal and pentadiagonal matrices. We refer to the matrices in the two classes as Toeplitz-regularized and corner-overlapped block-diagonal matrices. The first class generalizes the tridiagonal Toeplitz matrices to matrices with larger bandwidths where boundary entries are modified. For the second class, we decompose the problem into a lower two-block matrix problem where one of the blocks is a quadratic eigenvalue problem (QEVP). Analytical solutions are also obtained for these QEVPs. Moreover, we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for MEVPs) for GMEVPs and derive some trigonometric identities. Possible generalizations and applications to the design of better numerical methods for solving partial differential equations are discussed.
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