Powers of Complex Persymmetric Antitridiagonal Matrices with Constant Antidiagonals

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). 1. Introduction Solving some difference, differential, and delay differential equations, we meet the necessity to compute the arbitrary positive integer powers of square matrix. Recently, computing the integer powers of antitridiagonal matrices has been a very popular problem. There have been several papers on computing the positive integer powers of various kinds of square matrices by Rimas et al., and others [1–5]. In 2011, the general expression for the entries of the power of complex persymmetric or skew-persymmetric antitridiagonal matrices with constant antidiagonals is presented by Gutiérrez-Gutiérrez [1]. Rimas [2] gave the general expression of the th power for this type of symmetric odd order antitridiagonal matrices ( ) in 2008. In [3, 4] a similar problem is solved for antitridiagonal 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 antitridiagonal matrices with zeros in main skew diagonal and elements ; in neighbouring diagonals is derived by Rimas [5]. In 2013, Rimas [6] gave the eigenvalue decomposition for real odd order skew-persymmetric antitridiagonal matrices with constant antidiagonals ( ) and derived the general expression for integer powers of such matrices. In the present paper, we derive a general expression for the th power of any complex persymmetric antitridiagonal matrices with constant antidiagonals ( ). This novel expression is both an extension of the one obtained by Rimas for the powers of the matrix with (see [2] for the odd case and [5] for the even case) and an extension of the one obtained by Honglin Wu for the powers of the matrix with (see [3] for the even case). 2. Derivation of General Expression In this present paper, we study the entries of positive integer power of an complex persymmetric antitridiagonal matrix with constant antidiagonals as follows: where ,？？ . Consider the following complex Toeplitz tridiagonal matrix: The next trivial result relates the matrix with and with the backward identity [1]: where is the Kronecker delta. Lemma 1. Let ,？？ , and . Then where and . Proof. We have This completes the proof. We will find the th power ( ) of the matrix (1). Theorem 2 relates all positive integer powers of the matrix