Explicit Expression for Arbitrary Positive Integer Powers of Special Sparse Matrices

Sparse matrices appear frequently in mathematical models. In this paper, we firstly present a general expression for the entries of the 
 
 r
 
 th 
 
 
 
 r
 ∈
 ℕ
 
 
 
 power of a certain 
 
 n
 
 -square sparse matrix, in terms of the Chebyshev polynomials of the second kind. Secondly, we present a method for integer positive powers of the skew matrix corresponding to these sparse matrices. This method will be inspiring to calculate the positive integer powers of the similar matrices. Finally, we present some examples to illustrate our results. Also, we give maple 18 procedures in order to verify our calculations.