Zero-dilation index of \(S_n\)-matrix and companion matrix (Q2834761)
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scientific article; zbMATH DE number 6655730
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Zero-dilation index of \(S_n\)-matrix and companion matrix |
scientific article; zbMATH DE number 6655730 |
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24 November 2016
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zero-dilation index
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\(S_n\)-matrix
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companion matrix
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numerical range
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spectral norm
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spectral radius
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rank
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0.92715216
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0.83852816
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0.8338387
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Zero-dilation index of \(S_n\)-matrix and companion matrix (English)
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The zero-dilation index \(d(A)\) of a complex \(n\times n\) matrix \(A\) is the largest integer \(k\) for which \(A\) is unitarily similar to a matrix with the \(k\times k\) zero submatrix in the left upper corner. The matrix \(A\) is an \(S_n\)-matrix if its spectral norm \(\|A\|\leq 1\), spectral radius \(\rho(A)<1\), and \(\text{rank}\,(I-A^*A)=1\). The authors prove that if \(A\) an \(S_n\)-matrix or a companion matrix, then \(d(A)\leq\lceil n/2\rceil\). They also give several equivalent conditions for equality.
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