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

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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