A proof of the \(T_n\) conjecture: Centralizers, Jacobians and spectrally arbitrary sign patterns (Q417540)
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scientific article; zbMATH DE number 6034505
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A proof of the \(T_n\) conjecture: Centralizers, Jacobians and spectrally arbitrary sign patterns |
scientific article; zbMATH DE number 6034505 |
Statements
A proof of the \(T_n\) conjecture: Centralizers, Jacobians and spectrally arbitrary sign patterns (English)
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14 May 2012
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centralizer
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Jacobian
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tridiagonal sign pattern
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characteristic polynomial
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0.85987043
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0.85473645
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0.8515897
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0.8461223
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0.84563375
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0.8454225
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This paper proves so-called \(T_n\) conjecture due to \textit{L. Elsner}, \textit{D. D. Olesky} and \textit{P. van den Driessche} [Linear Algebra Appl. 374, 219--230 (2003; Zbl 1033.15004)] which states that for every real monic polynomial \(p(x)\) of degree \(n\geq 2\) there exists an \(n\times n\) tridiagonal sign pattern NEWLINE\[NEWLINE \left[ \begin{matrix} - & + & 0 & \cdots & 0 \\ - & 0 & \ddots & & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & 0 & + \\ 0 &\cdots & 0 & - & + \end{matrix} \right] NEWLINE\]NEWLINE whose characteristic polynomial is \(p(x)\).
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