The eigenvalues of a tridiagonal matrix in biogeography

We derive the eigenvalues of a tridiagonal matrix with a special structure. A conjecture about the eigenvalues was presented in a previous paper, and here we prove the conjecture. The matrix structure that we consider has applications in biogeography theory. 2011 Elsevier Inc. All rights reserved. 1. Main result and related work We prove the following theorem in this paper. Theorem 1. The (n + 1) (n + 1) tridiagonal matrix A 1⁄4 1 1=n 0 0 n=n 1 2=n . . . .. .. . . . . . . . . . . .. . .. . . . . 2=n 1 n=n 0 0 1=n 1 2 66666664 3 77777775 ð1Þ for any natural nP 4 has the (n + 1) eigenvalues xðAÞ 1⁄4 f0; 2=n; 4=n; . . . ; 2g 1⁄4 2k=n; k 2 1⁄20;n : ð2Þ This theorem was stated as a conjecture in [14] and is proven here in Sections 2–4. This type of matrix arises in biogeography theory, and its application is discussed in Section 5. There have been many papers published on explicit solutions of eigensystems of tridiagonal matrices of different structures. Some research on this topic includes [2,5,7,10,11,16]. The common feature of [2,5,7,16] is that the tridiagonal matrix A has the following structure: 0096-3003/$ see front matter 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.05.054 ⇑ Corresponding author. E-mail address: [email protected] (D. Simon). Applied Mathematics and Computation 218 (2011) 195–201