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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.
MSC:
15A18 Eigenvalues, singular values, and eigenvectors
92D25 Population dynamics (general)
92D40 Ecology
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[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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