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A simple derivation of the eigenvalues of a tridiagonal matrix arising in biogeography. (English) Zbl 1329.15027
The Sylvester-Kac matrix, also known as Clement matrix, is the \((n+1)\times (n+1)\) tridiagonal matrix \[ A_n=\begin{pmatrix} 0 & 1 &&&& \\ n & 0 & 2 &&& \\ & n-1 & \ddots & \ddots && \\ && \ddots & \ddots & n-1 & \\ &&& 2 & 0 & n \\ &&&& 1 & 0 \\ \end{pmatrix}. \] This matrix has been independently rediscovered and extended by many authors with interesting applications (see, for example, the paper by the reviewer, D. A. Mazilu et al., [Appl. Math. Lett. 26, No. 12, 1206–1211 (2013; Zbl 1311.15016)] and references therein). Its eigenvalues are well known: \[ n-2k\, ,\; \text{for }k=0,\ldots,n. \] The matrix studied in this note and its spectrum were considered by B. Igelnik and D. Simon [Appl. Math. Comput. 218, No. 1, 195–201 (2011; Zbl 1255.15009)] and D. Simon [“Biogeography-based optimization”, IEEE Trans. Evolutionary Comput. 12, 702–713 (2008)]. This matrix is an immediate derivation of the Sylvester-Kac matrix and its spectrum is well known. It seems that neither the just mentioned authors nor the authors of the paper under review were aware of this fact.
15A18 Eigenvalues, singular values, and eigenvectors
92D25 Population dynamics (general)
92D40 Ecology
Full Text: DOI
[1] [1]B. Igelnik and D. Simon, The eigenvalues of a tridiagonal matrix in biogeography, Appl. Math. Comput. 218 (2011), 195–201. Eigenvalues of a tridiagonal matrix27 · Zbl 1255.15009
[2] [2]D. Simon, Biogeography-based optimization, IEEE Trans. Evolutionary Comput. 12 (2008), 702–713.
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