Eigenvalues of tridiagonal pseudo-Toeplitz matrices

Abstract

In this article we determine the eigenvalues of sequences of tridiagonal matrices that contain a Toeplitz matrix in the upper left block.

Dynamic fluid sloshing in a one-dimensional array of coupled vessels

“…The eigenvectors of a tri-diagonal-Toeplitz matrix are known exactly [27,28], and thus lead to the values of the amplitude coefficients for the j th root of c 2 given in (4.6). For the specific cases in figure 8 these give the corresponding eigenvectors as…”

“…is the determinant of an N × N tri-diagonal-Toeplitz matrix with G 1 + ξ in every element along the leading diagonal, and −G 1 in every element along the lower and upper diagonals [27,28]. Note that A 0 = 1.…”

Transient solutions for multi-server queues with finite buffers

“…. , g c recursively using (12) and (13). Given g c−1 and g c we can find g n for all n > c by using (14), which is a homogeneous difference equation.…”

“…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].…”

“…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]. We must also mention the work of van Doorn [20] and Karlin and McGregor [9].…”