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The interesting spectral interlacing property for a certain tridiagonal matrix. (English) Zbl 1450.15009

A Sylvester-Kac matrix \(A_n\) stands for the following tridiagonal matrix: \[ A_n=\begin{pmatrix} 0& 1 & 0& \cdots & 0\\ n & 0 & 2 & &\vdots \\0 & n-1 &\ddots &\ddots & 0\\ \vdots & & \ddots & 0& n\\ 0& \cdots &0 & 1& 0 \end{pmatrix}.\]
The authors present a special matrix \(H_n\) (similar) to \(A_n\), \[H_n=\begin{pmatrix}\\ 0&\frac{1}{2} & & & & \\n&0&\frac{2}{2}& & & \\&\frac{2n-1}{2} &\ddots & \ddots&& \\& &\ddots & \ddots&\frac{n-1}{2}& \\& & & \frac{n+2}{2}&0&n\\& & & & \frac{n+1}{2}&0 \end{pmatrix}\] having the same \(n\)-Sylvester spectrum \(\mathrm{Sp}(A_n)=\{n-2k, k=0,\ldots,n\}. \)
The idea is based on an indirect trigonalisation (see also C. M. da Fonseca et al., Appl. Math. Lett. 26, No. 12, 1206–1211 (2013; Zbl 1311.15016)]).

MSC:

15A20 Diagonalization, Jordan forms
15A18 Eigenvalues, singular values, and eigenvectors
15A15 Determinants, permanents, traces, other special matrix functions

Citations:

Zbl 1311.15016
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References:

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