Eigenvalues and eigenvectors of tridiagonal matrices

Abstract

Abstract. This paper is continuation of previous work by the present author, where explicit formulas for the eigenvalues associated with several tridiagonal matrices were given. In this paper the associated eigenvectors are calculated explicitly. As a consequence, a result obtained by WenChyuan Yueh and independently by S. Kouachi, concerning the eigenvalues and in particular the corresponding eigenvectors of tridiagonal matrices, is generalized. Expressions for the eigenvectors are obtained that differ completely from those obtained by Yueh. The techniques used herein are based on theory of recurrent sequences. The entries situated on each of the secondary diagonals are not necessary equal as was the case considered by Yueh. Key words. Eigenvectors, Tridiagonal matrices.AMS subject classifications. 15A18. Introduction.The subject of this paper is diagonalization of tridiagonal matrices. We generalize a result obtained in [5] concerning the eigenvalues and the corresponding eigenvectors of several tridiagonal matrices. We consider tridiagonal matrices of the formwhere {a j } n−1 j=1 and {c j } n−1 j=1 are two finite subsequences of the sequences {a j } ∞ j=1 and {c j } ∞ j=1 of the field of complex numbers C, respectively, and α, β and b are complex numbers. We suppose thatwhere d 1 and d 2 are complex numbers. We mention that matrices of the form (1) are of circulant type in the special case when α = β = a 1 = a 2 = ... = 0 and all the entries on the subdiagonal are equal. They are of Toeplitz type in the special case when α = β = 0 and all the entries on the subdiagonal are equal and those on the superdiagonal are also equal (see U. Grenander and G. Szego He has calculated, in this case, the eigenvalues and their corresponding eigenvectorswhere θ k = where d is a complex number. We have proved that the eigenvalues remain the same as in the case when the a i 's and the c i 's are equal but the components of the eigenvector u (k) (σ) associated to the eigenvalue λ k , which we denote by uwhere θ k is given by formula d 2 sin (n + 1) θ k − d (α + β) sin nθ k + αβ sin (n − 1) θ k = 0, k = 1, ..., n.Recently in S. Kouachi [6], we generalized the above results concerning the eigenvalues of tridiagonal matrices (1) satisfying condition (2), but we were unable to calculate the corresponding eigenvectors, in view of the complexity of their expressions. The

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“…Both spectral expressions generalize many particular cases known in the literature, as for example it can be seen in [5,7,[9][10][11][12]14,[28][29][30][31][32][33].…”

Transient solutions for multi-server queues with finite buffers

“…For similar results see [8,11,12,19,23]. Also useful is the following equation, which immediately follows from (23):…”

“…Similar solutions are given by Morse [14], Riordan [17], van Assche et al [19] and Wagner [21, p. 860], the last one being an introductory textbook for Operations Research. We should also mention the paper of Böttcher and Grundsky [1], Kulkarni et al [12], Kouachi [11] and Yueh [23], who analyzed eigenvalues of pseudo-Toeplitz matrices, which is a slightly different problem, but which has the same general solution. As is well known, the problem in question is closely related to Chebychev polynomials [12,19].…”

Analytic Results for the Eigenvalues of Certain Tridiagonal Matrices

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“…365-366]. Kouachi [6] used a different method to study eigenvalues and eigenvectors of the matrix A and generalized to the case where the off-diagonal entries satisfy…”