# Matrix Computations and Semiseparable Matrices: Eigenvalue by Raf Vandebril

By Raf Vandebril

The normal houses and mathematical constructions of semiseparable matrices have been awarded in quantity 1 of Matrix Computations and Semiseparable Matrices. In quantity 2, Raf Vandebril, Marc Van Barel, and Nicola Mastronardi talk about the speculation of established eigenvalue and singular price computations for semiseparable matrices. those matrices have hidden houses that permit the improvement of effective equipment and algorithms to correctly compute the matrix eigenvalues.

This thorough research of semiseparable matrices explains their theoretical underpinnings and includes a wealth of knowledge on imposing them in perform. a number of the exercises featured are coded in Matlab and will be downloaded from the net for extra exploration.

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Additional resources for Matrix Computations and Semiseparable Matrices: Eigenvalue and Singular Value Methods (Volume 2)

Example text

Let Ui be a Householder (l) (l) reﬂector, such that the product A0 H1 has the entries 1 up to n − l − 1 annihilated in the n − l + 1-th row, thereby changing the columns 1 up to n − l of the matrix. Step 1. We will start by making all the elements in the last row, except (1) (1) for the last two, zero. Just multiply A0 to the right by H1 to create the wanted zeros. To complete the similarity transformation, we also perform the (1) T (1) (1) transformation H1 to the left of A0 H1 . This last transformation on the left does not destroy the zeros created by the transformation on the right, as this transformation only interacts with rows 1 to n − 1.

Similarity transformations are useful if the aim is to compute the eigenvalues of the original matrix. If one wants to compute the singular values, however, there is no reason to stick to similarity transformations. Therefore, we propose in this section orthogonal transformations to come to an easy form, useful for computing the singular values. In correspondence with the reduction of an arbitrary matrix by orthogonal transformations to a bidiagonal one, we construct an algorithm that 19 20 Chapter 2.

Gohberg. On a new class of structured matrices. Integral Equations and Operator Theory, 34:293–324, 1999. ☞ E. E. Tyrtyshnikov. Mosaic ranks for weakly semiseparable matrices. In M. Griebel, S. Margenov, and P. Y. Yalamov, editors, Large-Scale Scientiﬁc Computations of Engineering and Environmental Problems II, volume 73 of Notes on Numerical Fluid Mechanics, pages 36–41. Vieweg, Braunschweig, Germany, 2000. In the ﬁrst article, the authors Eidelman and Gohberg investigate a generalization of the class of semiseparable matrices, namely, the class of quasiseparable matrices in its most general form.