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eigenvalues of matrices or operators
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List of analytically known eigensystems?
To start the list, tridiagonal matrices have elementary expressions for its eigenvalues and eigenvectors. See, for example, works related to Molinari et al. … Block multi-diagonal matrices also have analytic expressions for their eigenvalues, see Section 4 of http://repository.uwyo.edu/cgi/viewcontent.cgi?article=1600&context=ela. …