On computing of arbitrary positive integer powers for one type of odd order symmetric circulant matrices—II
Introduction
Solving some difference, differential equations and delay differential equations we meet the necessity to compute the arbitrary positive integer powers of square matrix [1], [2]. In the work [3] the general expression of the lth power (l ∈ N) for one type of symmetric odd order tridiagonal matrices is presented. In this new paper we give the complete derivation of the general expression, presented in [3].
Section snippets
Formulation of the problem
Consider the nth order (n = 2p + 1, p ∈ N) matrix B of the following type:
We will derive expression of the lth power (l ∈ N) of the matrix (1) applying the expression Bl = TJlT−1 [5], where J is the Jordan’s form of B, T is the transforming matrix. Matrices J and T can be found provided eigenvalues and eigenvectors of the matrix B are known. The eigenvalues of B are defined by the characteristic equationIn the paper (3) it is shown that the roots of the characteristic equation
Eigenvectors of matrix B and transforming matrix T
Consider the relation J = T−1BT (BT = TJ); here B is the nth order matrix (1) (n = 2p + 1, p ∈ N), J is the Jordan’s form of B, T is the transforming matrix. We will find the transforming matrix T.
Denoting jth column of T by , we have T = (T1T2…Tn) andThe latter expression gives:Denotingand solving the set of systems (5), we find
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