Milan Hladík. Description of symmetric and skew-symmetric solution set. SIAM J. Matrix Anal. Appl., 30(2):509–521, 2008.
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We consider a linear system $Ax=b$, where $A$ is varying inside a given interval matrix $A$, and $b$ is varying inside a given interval vector $b$. The solution set of such a system is described by the well-known Oettli-Prager Theorem. But if we are restricted only on symmetric / skew-symmetric matrices $A\in\mathbfA$, the problem is much more complicated. So far, the symmetric / skew-symmetric solution set description could be obtained only by a lengthy Fourier-Motzkin elimination applied on each orthant. We present an explicit necessary and sufficient characterization of the symmetric and skew-symmetric solution set by means of nonlinear inequalities. The number of the inequalities is, however, still exponential w.r.t. the problem dimension.
@article{Hla2008g,
author = "Milan Hlad\'{\i}k",
title = "Description of symmetric and skew-symmetric solution set",
journal = "SIAM J. Matrix Anal. Appl.",
fjournal = "SIAM Journal on Matrix Analysis and Applications",
volume = "30",
number = "2",
pages = "509-521",
year = "2008",
doi = "10.1137/070680783",
bib2html_dl_html = "http://link.aip.org/link/?SML/30/509",
bib2html_dl_pdf = "https://epubs.siam.org/doi/10.1137/070680783",
abstract = "We consider a linear system $Ax=b$, where $A$ is varying inside a given interval matrix $A$, and $b$ is varying inside a given interval vector $b$. The solution set of such a system is described by the well-known Oettli-Prager Theorem. But if we are restricted only on symmetric / skew-symmetric matrices $A\in\mathbf{A}$, the problem is much more complicated. So far, the symmetric / skew-symmetric solution set description could be obtained only by a lengthy Fourier-Motzkin elimination applied on each orthant. We present an explicit necessary and sufficient characterization of the symmetric and skew-symmetric solution set by means of nonlinear inequalities. The number of the inequalities is, however, still exponential w.r.t. the problem dimension.",
keywords = "linear interval systems, solution set, interval matrix, symmetric matrix",
}
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