eu ON TENSOR PRODUCT DECOMPOSITION OF k-TRIDIAGONAL TOEPLITZ MATRICES

In the present paper, we provide a decomposition of a k-tridiagonal Toeplitz matrix via tensor product. By the decomposition, the required memory of the matrix is reduced and the matrix is easily analyzed since we can use properties of tensor product.

A k-tridiagonal matrix is one of generalizations of a tridiagonal matrix and has received much attention in recent years (e.g., [3], [6], [9], and [10]).Here, let T (k) n be an n-by-n k-tridiagonal matrix defined as If n is a k-tridiagonal Toeplitz matrix.Moreover, when k = 1, T (1) n is an ordinary tridiagonal Toeplitz matrix.We consider a k-tridiagonal Toeplitz matrix Hereafter, tensor product is briefly explained since it is used in a decomposition of k-tridiagonal Toeplitz matrices in the present paper.Tensor product is also referred to as the Kronecker product and represented by the symbol "⊗".The definition of tensor product of matrices A ∈ C m×n and B ∈ C p×q is where a ij is the (i, j) element of A. Let âi and a j be the i-th row and the j-th column vectors in A, respectively.Similarly, let bi and b j be the i-th row and the j-th column vectors in B, respectively.Then, as the other expressions of A ⊗ B.
The purpose of the present paper is to save the required memory of a ktridiagonal Toeplitz matrix and to simplify analyses of that.We propose to decompose the k-tridiagonal Toeplitz matrix into a smaller matrix with the similar structure than the original one and an identity matrix via tensor product.Even if the number n in T (k) n is very large, the matrix is decomposed into T n by using properties of tensor product.This paper is organized as follows.In Section 2, we give a theorem of the decomposition via tensor product and show two examples.In Section 3, the decomposition is applied in order to simplify the theorem and the proposition in [13] and to reduce a computational complexity.

Main Results
In this section, we present a theorem of a decomposition of a k-tridiagonal Toeplitz matrix T n is decomposed into the form: where I m represents the identity matrix of order m.
Proof.First, let T and S be Toeplitz matrices of the same size.Then, T is equal to S if and only if both of the following equations are satisfied: (T ) 1: = (S) 1: for the first row vectors in T and S; (T ) :1 = (S) :1 for the first column vectors in those.Here, (T ) i: and (T ) :j denote the i-th row and the j-th column vectors in T , respectively. Since n ′ ⊗ I m also has Toeplitz structure.Hence, the two matrices are the same if both of the first column and row vectors in T (T (T where e 1 and 0 m represent the m-dimensional first canonical vector and the mdimensional zero vector, respectively.From ( 5) and ( 6), we have (T This completes the proof.
Theorem 1 provides three notes: first, the required memory of is the lowest in all the values m when m = gcd(n, k); second, the k ′ -tridiagonal Toeplitz matrix T n ′ under the condition that n = mk; third, the determinant, the eigenvalues, integer powers, and the inversion of T From Examples 2 and 3, we can confirm the first and second notes.Particularly, we can see that the number of nonzero elements of matrices is reduced by one m-th.

Applications
In Subsection 3.1, we present some corollaries, which are obtained from Theorem 1.Some of the corollaries imply the theorem and the proposition that were proved in [13], however the corollaries in the present paper have simpler and more general expressions than in [13].In Subsection 3.2, we show that Theorem 1 is used in order to reduce a computational complexity.

The Symmetric k-Tridiagonal Toeplitz Matrices
We consider a specialized form of symmetric k-tridiagonal Toeplitz matrices, i.e., where i, j = 1, 2, . . ., n, and parameters n and k are natural numbers such that n = mk.Applying Theorem 1 to S n in (7), we obtain Corollary 4.
n be the matrix as in (7).Then Proof.By the definition of S n and Theorem 1, the result can be obtained.
Using Corollary 4, the determinant, the eigenvalues, and arbitrary integer powers of S n be the matrix as in (7).Then Proof.By Corollary 4 and the property of the determinant of tensor product, we have n be the matrix as in (7).Then, the eigenvalues λ j of S Proof.The result is obtained by Corollary 4, the analytical forms of the eigenvalues of the tridiagonal Toeplitz matrix (cf.[8,Example 7.2.5]), and the eigenvalues of tensor product.
Note that the eigenvalues are particular forms of those in [7].

Corollary 7. Let S (k)
n be the matrix as in (7).Then where r is an arbitrary integer.
Proof.By Corollary 4 and the property of tensor product, the result is obvious.

A Reduction of a Computational Complexity by the Decomposition
Theorem 1 can be used in order to reduce the computational complexity.We here focus on the computation of the inversion of the k-tridiagonal Toeplitz matrix T has elements, the lower the computational complexity of the algorithm is.Since the number of nonzero elements of T is reduced by one m-th, the computational complexity is also reduced by one m-th.

Conclusion
In the present paper, we gave a decomposition of a k-tridiagonal Toeplitz matrix via tensor product.As applications of the decomposition, we have shown that the determinant, the eigenvalues, and arbitrary integer powers of the matrix are easily computed and that the inversion of the matrix is computed with lower computational complexity than that without the decomposition.
matrix under a certain condition.Then, one needs only analyses of T

n
and show two examples.First, the theorem is as follows: Theorem 1.Let T (k) n be an n-by-n k-tridiagonal Toeplitz matrix.If there exist natural numbers n ′ , k ′ , and m such that n = mn ′ and k = mk ′ , where m > 1, T

n
and in T (k ′ ) n ′ ⊗ I m are equal.From (3), the first row and column vectors are obtained as follows:

2 ⊗ 4 ⊗
from those of T (k ′ ) n ′ .As for the condition in the second note, the original matrix T (k) n is decomposed into the tridiagonal Toeplitz matrix of order 2 and the identity matrix of order k, i.e., T I k , under the condition that n = 2k.Next, two examples under the condition that n = mk are shown.Example 2. Let n = 8 and k = 2. Setting m = 2, then the 2-tridiagonal Toeplitz matrix T (2) 8 is decomposed such as T I 2 .The matrices are specifically denoted as follows:

n
are easily computed as below.Corollary 5. Let S (k)

using [ 5 ,
Theorem 2.1].An algorithm to compute the inversion with the decomposition is shown below.First, the k-tridiagonal Toeplitz matrixT (k)n is decomposed by Theorem 1.Then, the inversion ofT (k) n is computed by (T (k) n ) −1 = (T (k ′ ) n ′ ) −1 ⊗ I m .Here, the algorithm in [5, Theorem 2.1] computes the inversion element-wise.Therefore, the fewer T (k) n