POWERS OF COMPLEX PERSYMMETRIC ANTI-TRIDIAGONAL MATRICES WITH CONSTANT ANTI-DIAGONALS

We derive a general expression for the pth power of any complex persymmetric antitridiagonal Hankel (constant antidiagonals) matrices. Numerical examples are presented, which show that our results generalize the results in the related literature (Rimas 2008, Wu 2010, and Rimas 2009).


INTRODUCTION
From a practical point of view, Anti-tridiagonal matrices arise frequently in many areas of mathematics and engineering, such as numerical analysis, solution of the boundary value problems, high order harmonic filtering theory [1] [2]. In many of such problems, we need to calculate some matrix functions such the powers, inverse or the exponential. There is a lot of work dealing with the inverse of a Anti-tridiagonal matrix and solving the resulting linear system can be done in an efficient way. However, computing the integer powers of Anti-tridiagonal matrices has been a very popular problem in last few years. There have been several papers on computing the positive integer powers of various kinds of square matrices by J.Rimas, Jes u s Guti e rrez, Yin, etc [3]- [7]. J.Rimas [4] gave the general expression of the p th power for this type of symmetric odd order anti-tridiagonal matrices ( (1,0,1) n g antitridia ) in 2008. In [5]- [6] a similar problem was solved for anti-tridiagonal matrices having zeros in main skew diagonal and units in the neighbouring diagonals. In 2010, the general expression for the entries of the power of odd order anti-tridiagonal matrices with zeros in main skew diagonal and elements 1 , 1, , 1 ,1; 1, 1,      in neighbouring diagonals was derived by J.Rams [7]. In 2011, the general expression for the entries of the power of complex persymmrtric or skew-persymmetric anti-tridiagonal matrices with constant anti-diagonals are presented by Jes u s Guti e rrez [3]. In 2013, J.Rimas [10] gave the eigenvalue decomposition for real odd order skew-persymmetric anti-tridiagonal matrices with constant anti-diagonals ( ) , , ( a b a g antitridia n ) and derived the general expression for integer powers of such matrices.
In the present paper, we derive a general expression for the p th power ) ( This paper is organized as follows: -In Section 2, we give the derivation of general expression. -In Section 3, Numerical examples are presented.
-In Section 3, we summarize the paper.

DERIVATION OF GENERAL EXPRESSION
In this paper, we study the entries of positive integer power of n n complex persymmetric anti-tridiagonal matrices with constant anti-diagonals (2.1) and where  is the Kronecker delta.
This completes the proof. We shall find the q th power ( N q  ) of the matrices (2.1) and (2. ,0, Proof. We will proceed by induction on q . The case 1 = q is obvious.
Suppose that the result is true for 1  q , and consider that case By the induction hypothesis we have With the tridiagonal matrix n D , we associate the polynomial sequence   characterized by a three-term recurrence relation: With initial conditions Proof. See [8]. Tending y to x in formulas (2.14) and (2.15), we get: ).
Since the matrix n A has distinct eigenvalues T is the transforming matrix formed by the eigenvectors of n A . Namely, are defined as above. For and S is the transforming matrix formed by the eigenvectors of n D . Namely, are defined as above.
where X is a closed curve containing the roots of n P and no roots of By using the Cauchy Integral Formula, we can give another expressions of the coefficients where X is a closed curve containing the roots of n P and no roots of By using the Cauchy Integral Formula, we can give another expressions of the coefficients

NUMERICAL EXAMPLES (a). The Persymmetric Case
Consider the n order anti-tridiagonal matrix n B of the following type: