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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In the current manuscript we comment on (Misra and Babu, Model Earth Syst Environ 2(1):1-11, 2016), where two novel five-species ODE models are proposed and analyzed, in order to investigate the population dynamics of a three-species food chain, in a polluted environment. It is shown in Misra and Babu (Model Earth Syst Environ 2(1):1-11, 2016) that under certain restrictions on the parameters, the models have bounded solutions for all positive initial conditions. Furthermore, a globally attracting set is explicitly constructed for initial conditions in R 5 þ. We prove these results are not true. To the contrary, solutions to these models can blow-up in finite time, even under the parametric restrictions derived in Misra and Babu (Model Earth Syst Environ 2(1):1-11, 2016), for sufficiently large initial conditions. We provide both analytical proofs and numerics to confirm our results.

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