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 .
It is well-known that the finite difference discretization of the Laplacian eigenvalue problem -\Delta u = \lambda u leads to a matrix eigenvalue problem (EVP) A x= \lambda x where the matrix A is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in Strang and MacNamara \mbox{%DIFAUXCMD strang2014functions analytical solutions to the generalized matrix eigenvalue problems ( GEVPs) A x= \lambda Bx which arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation . Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly , we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities .
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We present
It is well-known that the finite difference discretization of the Laplacian eigenvalue problem -\Delta u = \lambda u leads to a matrix eigenvalue problem (EVP) A x= \lambda x where the matrix A is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in Strang and MacNamara \mbox{%DIFAUXCMD strang2014functions
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two classes of
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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
GEVPs) A x= \lambda Bx which arise from the finite element method (FEM)
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isogeometric analysis (IGA). The FEM matrices are
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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)
while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation
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these QEVPs. Moreover
some polynomial eigenvalue problems (PEVPs). Lastly
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MEVPs) for GMEVPs
EVPs) for GEVPs
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. Possible generalizations and applications to the design of better numerical methods for solving partial differential equations are discussed