Volume 36, pp. 99-112, 2009-2010.

The structured distance to nearly normal matrices

Laura Smithies

Abstract

In this note we examine the algebraic variety ${\cal I}_\Lambda$ of complex tridiagonal $n \times n$ matrices $T$, such that $T^*T - T T^* = \Lambda$, where $\Lambda$ is a fixed real diagonal matrix. If $\Lambda = {\bf 0}$ then ${\cal I}_\Lambda$ is ${\cal N}_{\bf T}$, the set of tridiagonal normal matrices. For $\Lambda \neq {\bf 0}$, we identify the structure of the matrices in ${\cal I}_\Lambda$ and analyze the suitability for eigenvalue estimation using normal matrices for elements of ${\cal I}_\Lambda$. We also compute the Frobenius norm of elements of ${\cal I}_\Lambda$, describe the algebraic subvariety ${\cal M}_\Lambda$ consisting of elements of ${\cal I}_\Lambda$ with minimal Frobenius norm, and calculate the distance from a given complex tridiagonal matrix to ${\cal I}_\Lambda$.

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Key words

nearness to normality, tridiagonal matrix, Kreǐn spaces, eigenvalue estimation, Geršgorin type sets

AMS subject classifications

65F30, 65F35, 15A57, 15A18, 47A25

Links to the cited ETNA articles

[2]Vol. 26 (2007), pp. 320-329 Natacha Fontes, Janice Kover, Laura Smithies, and Richard S. Varga: Singular value decomposition normally estimated Geršgorin sets
[5]Vol. 28 (2007-2008), pp. 65-77 S. Noschese, L. Pasquini, and L. Reichel: The structured distance to normality of an irreducible real tridiagonal matrix

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