ON INVERSES AND EIGENPAIRS OF PERIODIC TRIDIAGONAL TOEPLITZ MATRICES WITH PERTURBED CORNERS∗

In this paper, we derive explicit determinants, inverses and eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners of Type I. The Mersenne numbers play an important role in these explicit formulas derived. Our main approaches include clever uses of the Schur complement and matrix decomposition with the Sherman-Morrison-Woodbury formula. Besides, the properties of Type II matrix can be also obtained, which benefits from the relation between Type I and II matrices. Lastly, we give three algorithms for these basic quantities and analyze them to illustrate our theoretical results.


Introduction
The main research object of this paper is an n × n matrix A = (a i,j ) n i,j=1 , which is called a periodic tridiagonal Toeplitz matrix with perturbed corners of type I and defined as follows where α 1 , α n , γ 1 , γ n , β are complex numbers with β = 0. LetÎ n be the n×n "reverse unit matrix", which has ones along the secondary diagonal and zeros elsewhere. Let A be defined as in (1.1). A matrix of the form B :=Î n AÎ n is called a periodic tridiagonal Toeplitz matrix with perturbed corners of type II. In this case, we say B is induced by A. It is readily seen that if A is a periodic tridiagonal Toeplitz matrix with perturbed corners of type I if and only if its transpose A T is a periodic tridiagonal Toeplitz matrix with perturbed corners of type II. General tridiagonal matrices appear not only in pure linear algebra, but also in many practical applications, such as, computer graphics [1], image denoising [26] and partial differential equations [9,34,35,40], etc. One takes the one-dimensional linear hyperbolic equation considered by Holmgren and Otto [13] as an example to study certain matrices occur in discretized partial differential equations, where 0 < x ≤ 1, t > 0, u(0, t) = f (−at), u(x, 0) = f (x), g = (v − a)f . Here v and a are positive constants and f is a scalar function with derivative f . Let k and h denote the time step and spatial step respectively. The linear hyperbolic equations discretized based on trapezoidal rule in time and center difference in space, respectively, whose coefficient matrix is a tridiagonal matrix with perturbed last row [2]: where α = vk/h. On the other hand, some parallel computing algorithms are also designed for solving tridiagonal systems on graphics processing unit (GPU), which are parallel cyclic reduction [14] and partition methods [39]. Recently, Yang et al. [41] presented a parallel solving method which mixes direct and iterative methods for block-tridiagonal equations on CPU-GPU heterogeneous computing systems, while Myllykoski et al. [27] proposed a generalized graphics processing unit implementation of partial solution variant of the cyclic reduction (PSCR) method to solve certain types of separable block tridiagonal linear systems. Compared to an equivalent CPU implementation that utilizes a single CPU core, PSCR method indicated up to 24-fold speedups. Many studies have been conducted for tridiagonal matrices [10,[16][17][18][19]43]. Typical results for their inverses include Usmani's algorithm [38] based on rudimentary matrix analysis, El-Mikkawy and Atlan's two symbolic algorithms [5,6] based on the Doolittle LU factorization of the k-tridiagonal matrix, Jia et al.'s algorithms [20,21] based on block diagonalization technique, and so on. There are also some studies on the inverse of nonsingular periodic tridiagonal matrices [7,22]. Tim and Emrah [37] used backward continued fractions to derive the LU factorization of periodic tridiagonal matrix and then derived the explicit formula for its inverse. Dow [4] discussed some special Toeplitz matrices including periodic tridiagonal Toeplitz matrices, while Shehawey [8] generalized Huang and McColl's [15] work and put forward the inverse formula for periodic tridiagonal Toeplitz matrices. Furthermore, some authors have done some research on the eigenpairs of tridiagonal matrices or periodic tridiagonal matrices based on the method of symbolic calculus for difference equations [3,11,42].
The rest of the paper is organized as follows: Section 2 illustrates the importance of the Mesenne numbers in the main results. Section 3 describes the detailed derivations of the determinants, inverses and eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners of Type I. Specifically, the formulas on representation of the determinants and inverses of these typies matrices in the form of products of Mersenne numbers and some initial values. Our main approaches include clever uses of the Schur complement [44, p.10] and matrix decomposition with Sherman-Morrison-Woodbury formula [12]. Besides, we calculate the eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners based on the eigenpairs of the symmetric tridiagonal Toeplitz matrix [23]. Furthermore, the properties of the periodic tridiagonal Toeplitz matrices with perturbed corners of Type II can be also obtained. Section 4 presents three algorithms for these basic quantities and analyze them to illustrate our theoretical results. The final conclusions are given in Section 5.

Mersenne Numbers and Applications
In this section, we introduce the Mersenne number M n , which satisfies the following recurrence [32]: It is known that the nth Mersenne number has the Binet formula M n = 2 n − 1.
Mersenne numbers are ubiquitous in combinatorics, number theory, group theory, chaos, geometry, physics, etc [25]. More specifically, Mersenne numbers play an important role in digital signal processing, which stems from arithmetic operations modulo Mersenne numbers can be implemented relatively simply in digital hardware [28]. Especially, Mersenne Number Transforms are often used to deal with problems of digital filtering and convolution of discrete signals [31,36]. For example, Nussbaumer proposed some digital filtering using pseudo-Mersenne transforms in [29] and pseudo-Fermat number transforms in [30].
It is clearly seen that the elements in the diagonal and superdiagonal of matrix A satisfy the recursive relationship of the Mersenne sequence. Those elements chosen in the diagonal and superdiagonal reflect the neat applications of Mersenne numbers. Besides, the structure of the matrix itself determines some basic quantities of the matrix such as determinant and inverse. Therefore, the Mersenne numbers play a huge role in our main results.

The Determinants, Inverses and Eigenpairs Formulas of Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners
In this section, we derive explicit formulas for the determinants, eigenpairs and inverses of periodic tridiagonal Toeplitz matrix with perturbed corners. Main effort is made for working out those for periodic tridiagonal Toeplitz matrix with perturbed corners of type I, since the results for type II matrices would follow immediately.
Proof. Define a circulant matrix Clearly, ρ is invertible and Multiply A by ρ from right and then partition Aρ into four blocks: and M i is the ith Mersenne number. Next, taking the determinants for both sides of (3.4) and by [44, p.10], we get To find det A, we need to evaluate the determinant of ( Finally, applying (3.3), (3.5) and (3.8) to (3.7), we get the determinant of A, which completes the proof.
be given as in (1.1) and assume A to be nonsingular. Then

9)
and For nonsingular A, we get from A −1 A = I that Similarly, according to AA −1 = I, we get a 1,j α n +ȃ n−1,j β +ȃ n,j γ n , 1 ≤ j ≤ n, i = n. (3.12) Based on (3.10), we get from (3.11) that Similarly, from (3.12), we get that (3.14) Based on the above analysis, we need to determine six initial values, that is, a i,j (i ∈ {1, 2, 3}, j ∈ {1, 2}), for the recurrence relations (3.13) and (3.14) in order to compute the inverse of A. The rest of the proof is devoted to evaluating these particular entries of A −1 .

It could be verified that ∆
and M −m is given in (2.2) for m = 1, 2, . . ..
Applying the Sherman-Morrison-Woodbury formula [12, p.50] to (3.15) gives Now we compute each component on the right side of (3.16).
Then multiplying (3.18) by K β from the left and further adding I, we have Computing the inverse of the matrices on both sides of (3.19), we obtain Multiplying the previous formula I + 1 β K∆ −1 L −1 by ∆ −1 L from the left and by K∆ −1 from the right, respectively, yields From (3.16) and (3.20), we have By (3.22), we compute, By (3.23), we compute, This completes the proof.
Remark 3.1. The formulas (3.22) and (3.23) would give an analytic formula for A −1 . However, there is a big advantage of (3.9) from computational consideration as we shall see from Section 3.

24)
and ϑ j = ρφη j , j = 1, 2, · · · , n, (3.25) where ρ is the same as in (3.2), , η j = (η 1,j , · · · , η n,j ) T , and consider the similarity transformations of A, where ρ is the same as in (3.2). Since similarity transformation preserves eigenvalues, it suffices to work out the eigenvalues of τ . It is easy to see the eigenvalues of τ are the union of the eigenvalues of A 11 and A 22 .
As the determinant of a complex matrix is equal to the product of its eigenvalues, we have the following corollary.
The next three theorems are parallel results of type I matrices. Proof. Since det B = detÎ n det A detÎ n , we obtain this conclusion by using Theorem 4 and detÎ n = (−1) Theorem 3.5. Let A be given as in (1.1) and let B be a periodic tridiagonal Toeplitz matrix with perturbed corners of type II, which is induced by A. Then Proof. It follows immediately from B −1 =Î −1 n A −1Î −1 n =Î n A −1Î n and Theorem 3.2.
Theorem 3.6. Let A be given as in (1.1) and let B be a periodic tridiagonal Toeplitz matrix with perturbed corners of type II, which is induced by A. Assume the eigenpairs of B are (λ j ,θ j ), j = 1, . . . , n. Then λ j is the same as in (3.24) and ϑ j =Î n ϑ j , where ϑ j is the same as in (3.25).
Proof. Since B is similar to A, B and A have the same eigenvalues λ j (j = 1, 2, · · · , n), where λ j is same with (3.24). The other claim is straightforward.

Algorithms
In this section, we give three algorithms for finding the determinant, inverse and eigenpairs of periodic tridiagonal Toeplitz matrix with perturbed corners of type I, which is called A. Besides, we make some analysis about these algorithms to illustrate our theoretical results. Firstly, based on Theorem 3.1, we give an algorithm for computing determinant of A as following: Step 1: Input α 1 , α n , γ 1 , γ n , β, order n and generate Mersenne number M n−1 by (2.1).
Step 2: Calculate and output the determinant of A by (3.1).
Based on Algorithm 1, we make a comparison of the total number operations for determinant of A between LU decomposition and Algorithm 1 in Table 1. Specifically, we get that the total number operation for the determinant of A is 2n + 1. What's more, this number can be reduced to O(logn) (see, [33], p.226-227).
Lastly, Algorithm 3 gives the eigenpairs of A based on Theorem 3.3. The total number operation for computing eigenpairs of A is 5n 2 + 20n − 23.

Conclusions
In this paper, we present the explicit formulas for determinants, inverses and eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners. The representation of the determinant in the form of products of Mersenne number and some initial values from matrix transformations. For inverse, our main approach includes a clever use of matrix decomposition with the Sherman-Morrison-Woodbury formula. Besides, we calculate the eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners based on the eigenpairs of the symmetric tridiagonal Toeplitz matrix. To test our method's effectiveness, we propose three algorithms for finding the determinants, inverses and eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners as well as compare the total number operation for these basic quantities between different algorithms. After comparison, we draw a conclusion that our algorithms are superior to other algorithms to some extent.