The square root of tridiagonal Toeplitz matrices

In this paper, we present an explicit formula to find square roots of a tridiagonal Toeplitz matrix, and we show that these square roots have the form of a persymmetric matrix with examples to illustrate.


Introduction
*The tridiagonal Toeplitz matrix is illustrated below: This type of matrix is used in several different fields of applications, such as a solution of ordinary and partial differential equations, time series analysis, and regularization matrices in Tikhonov regularization for the solution of discrete ill-posed problems. It is, therefore, important to understand the properties of this type of matrix (Noschese et al., 2013). Yuttanan and Nilrat (2005) gave an answer to the question of which matrices have an ℎ root for any positive integer and which have an ℎ root only for some positive integer . As a special case the diagonalizable matrices always have the ℎ roots.
In Section 2 and through the work of Salkuyeh (2006) about positive integer powers of the tridiagonal Toeplitz matrices, we give a method to calculate square roots of this type of matrix with proof the form of the square root in section 3 with examples to illustrate.

The square root of tridiagonal Toeplitz matrices
In this section, we will give an explicit statement to the square root of the tridiagonal Toeplitz matrix, and we start with present some important results.
Theorem 1: Let B be a complex matrix of order m. If B is diagonalizable, then B has an ℎ root for any positive integer n.
As a special case, the matrix has a square root, i.e., there exists a matrix such that: 2 = (Yuttanan and Nilrat, 2005).
Lemma 1: Let A be a tridiagonal Toeplitz matrix defined in (1.1), the eigenvalues and eigenvectors of A are given by, Moreover, the matrix is diagonalizable, i.e., there exists an invertible matrix such that: (2000).

Structure of square root of tridiagonal Toeplitz matrices
In this section, we prove that the square root of the tridiagonal Toeplitz matrix defined in (2.8) takes the form of a persymmetric matrix.
Let's start with the definition of a persymmetric matrix.

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Theorem 3: The square root R = (r i j ) of tridiagonal Toeplitz matrices defined in (2.8) is a persymmetric matrix: Proof: To prove (3.1), you must note the following: Now by using (2.8), we have: Observe that is a persymmetric matrix.

Conclusion
Based on this article, we obtain a formula for calculating a square root of a tridiagonal Toeplitz matrix, which is diagonalizable. The reader can find other properties and other square roots of other diagonalizable matrices with the same method we used.