Episode March 2, 2026 14 min 54s

Matrix Computations and Semiseparable Matrices: Eigenvalue and Singular Value Methods

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Explores various matrix computations, with a particular focus on semiseparable matrices and their applications. It defines semiseparable matrices, details their properties, and discusses different representations, such as generator and Givens-vector forms. The text covers algorithms for reducing matrices into semiseparable, tridiagonal, and Hessenberg forms, utilizing transformations like Givens and Householder rotations. Furthermore, it examines the convergence properties of these reduction algorithms, relates them to Lanczos and Arnoldi-Ritz values, and discusses the implementation and numerical experiments of QR-algorithms for eigenvalue and singular value computations of structured matrices, including techniques for deflation and eigenvector calculation. Finally, the source touches upon inverse eigenvalue problems and orthogonal polynomials in the context of structured matrix computations.

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