## Qualitative Analysis of Differential, Difference Equations, and Dynamic Equations on Time Scales

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# On the Minimal Polynomials and the Inverses of Multilevel Scaled Factor Circulant Matrices

**Academic Editor:**Tongxing Li

#### Abstract

Circulant matrices have important applications in solving various differential equations. The level-*k* scaled factor circulant matrix over any field is introduced. Algorithms for finding the minimal polynomial of this kind of matrices over any field are presented by means of the algorithm for the Gröbner basis of the ideal in the polynomial ring. And two algorithms for finding the inverses of such matrices are also presented. Finally, an algorithm for computing the inverse of partitioned matrix with level-*k* scaled factor circulant matrix blocks over any field is given by using the Schur complement, which can be realized by CoCoA 4.0, an algebraic system, over the field of rational numbers or the field of residue classes of modulo prime number.

#### 1. Introduction

Circulant matrices play an important role in solving many different differential equations, such as ordinary, partial, matrix, linear second-order partial, bi-Hamiltonian partial, parameterized delay, fractional order, and singular perturbation delay. Lee et al. investigated a high-order compact (HOC) scheme for the general two-dimensional (2D) linear partial differential equation in [1] with a mixed derivative. Meanwhile, in order to establish the CCD2 scheme, they rewrote equation (1.1) into (2.1) in [1]. To write the CCD2 system in a concise style, they employed circulant matrix to obtain the corresponding whole CCD2 linear system (2.10), whose entries are circulant block. Using circulant matrix, Karasözen and Şimşek [2] considered periodic boundary conditions such that no additional boundary terms will appear after semidiscretization. Guo et al. concerned generic Dn-Hopf bifurcation to a delayed Hopfield-Cohen-Grossberg model of neural networks (5.17) in [3], where denoted an interconnection matrix. In particular, they assumed that is a symmetric circulant matrix. Trench considered nonautonomous systems of linear differential equations (1) in [4] with some constraints on the coefficient matrix . One case is that the is a variable block circulant matrix. In [5], some Routh-Hurwitz stability conditions are generalized to the fractional order case. Ahmed et al. considered the 1-system CML (10) in [5]. They selected a circulant matrix, which reads a tridiagonal matrix. In [6], Jin et al. proposed the GMRES method with the Strang-type block-circulant preconditioner for solving singular perturbation delay differential equations. In [7], Claeyssen and Leal introduce factor circulant matrices: matrices with the structure of circulants, but with the entries below the diagonal multiplied by the same factor. The diagonalization of a circulant matrix and spectral decomposition are conveniently generalized to block matrices with the structure of factor circulants. Matrix and partial differential equations involving factor circulants are considered. Wilde [8] developed a theory for the solution of ordinary and partial differential equations whose structure involves the algebra of circulants. He showed how the algebra of circulants is related to the study of the harmonic oscillator, Cauchy-Riemann equations, Laplace’s equation, the Lorentz transformation, and the wave equation. And he used circulants to suggest natural generalizations of these equations to higher dimensions.

With the development of the mathematical research, multilevel circulant matrix had been defined. And it has been used on network engineering, approximate calculation, and Image processing [9–12]. Jiang and Liu [13] introduced the level- scaled circulant factor matrix over the complex number field and discussed its diagonalization and spectral decomposition and representation. Zhang et al. [14] gave algorithms for the minimal polynomial and the inverse of a level--block circulant matrix over any field by means of the algorithm for the Gröbner basis for the ideal of the polynomial ring over the field. Morhac and Matousek [15] present an efficient algorithm to solve a one-dimensional as well as -dimensional circulant convolution system. Rezghi and Elden [16] defined tensors with diagonal and circulant structure and developed a framework for the analysis of such tensors. Georgiou and Koukouvinos [17] presented a new method for constructing multilevel supersaturated designs. Trench [18, 19] considered properties of unilevel block circulants and multilevel block -circulants. Block [20] considered the property of circulants of level-. Baker et al. discussed the structure of multiblock circulants in [21]. More details on multilevel circulant matrix can be found in [22–24].

This paper is devoted to study the level- scaled factor circulant matrix, and it is organized as follows.

In Section 2, a level- scaled factor circulant matrix over any field is introduced and its algebraic properties are given.

In Section 3, we first show that the ring of all level- scaled factor circulant matrices over a field is isomorphic to a factor ring of a polynomial ring in variables over the same field, and then we present an algorithm for finding the minimal polynomial of a level- scaled factor circulant matrix by mean of the algorithm for the Gröbner basis for a kernel of a ring homomorphism.

In Section 4, we give a sufficient and necessary condition to determine whether a level- scaled factor circulant matrix over a field is singular or not and then present an algorithm for finding the inverse of such a matrix over a field.

In Section 5, an algorithm for finding the inverse of partitioned matrix with level- scaled factor circulant matrix blocks over a field is presented by using the Schur complement and Buchberger’s algorithm.

We first introduce some terminologies and notations used in the equations. Let be a field and the polynomial ring of variables over field . By Hilbert basis Theorem, we know that every ideal in is finitely generated. Fixing a term order in , a set of nonzero polynomials in an ideal is called a Gröbner basis for if and only if, for all nonzero , there exists such that divides , where and are the leading power products of and , respectively. A Gröbner basis is called a reduced Gröbner basis if and only if, for all and is reduced with respect to ; that is, for all , no non-zero term in is divisible by any for any , where is the leading coefficient of .

In this paper, we set for a square matrix , and denotes an ideal of generated by polynomials .

#### 2. Level- Scaled Factor Circulant Matrices

If is an matrix over field which is the product of a diagonal matrix and a circulant permutation matrix , this is

Then, the matrix is called a scaled circulant permutation matrix over field .

When field is the complex field, this kind of matrix is the same as in [25].

For the remainder of the paper, the indices are congruence classes modulo . We will use instead of . For convenience, we will refer to such a matrix as an SCPMF.

As is a scaled circulant permutation matrix over field , then

In this paper, focus on the case where is nonsingular SCPMF, where It is easy to show that the polynomial is both the minimal polynomial and the characteristic polynomial of .

Let be the unit matrix for and . Set where is a Kronecker product of matrices.

*Definition 1. *An maxtrix over is callled a level- scaled factor circulant matrix if there exists a polynomial
such that
where will be called the representer of a level- scaled factor circulant matrix .

Obviously, when field is the complex field and , this kind of matrix is same as in [25], and when the field is the complex field, this kind of matrix is the same as in [13], and if , , this kind of matrix is as in [14, 18, 22], and if , , then we obtain the multilevel circulant matrix [9–12, 15, 19–22].

From the property of the Kronecker product of matrices, the level- scaled factor circulant matrix can also be expressed as

For a matrix over , is a level- scaled factor circulant matrix if and only if commutes with ; that is,

In addition to the algebraic properties that can be easily derived from representation (6), we mention that level- scaled factor circulant matrices have very nice structure. The product of two level- scaled factor circulant matrices is also a level- scaled factor circulant matrix. Furthermore, level- scaled factor circulant matrices commute under multiplication and is also a level- scaled factor circulant matrix.

#### 3. Minimal Polynomials of Level- Scaled Factor Circulant Matrices

Let . It is a routine to prove that is a commutative ring with the matrix addition and multiplication.

Theorem 2. *Consider .*

*Proof. *Consider the following -algebra homomorphism:
for . It is clear that is an -algebra epimorphism. So, we have

We can prove that
In fact, for , because . Hence, .

Conversely, for any , we have . Fix the lexicographical order on with . Consider dividing , and there exist
such that
where or the largest degree of in is less than . If , then . Otherwise, dividing , and there exist , such that
where or the largest degree of in is less than . If , then . Otherwise, the largest degree of in is less than because does not appear in . Continuing this procedure, there exist , and , such that , where or the degrees of in are less than , respectively. Since . For . The coefficients of all terms in are the entries of the matrix because the degrees of in are less than , respectively. Therefore, the coefficient of each term in is ; that is, . Thus,

*Definition 3. *Let be a nonzero ideal of the polynomial ring . Then, is called an annihilation ideal of square matrices , denoted by , if for all .

*Definition 4. *Suppose that are not all zero matrices. The unique monic polynomial of minimum degree that simultaneously annihilates is called the common minimal polynomial of .

We give the special case of Theorem [26] here for the convenience of applications.

Lemma 5. * Let be an ideal of . Given , consider the following F-algebra homomorphism:
**
Let be an ideal of generated by . Then, .*

The following lemma is well known [27].

Lemma 6. * Let be a nonzero matrix over field . If the minimal polynomial of is
**
then
*

The following lemma is the Exercise 2.38 of [26].

Lemma 7. * Let be ideals of and let be an ideal of generated by . Then, .*

By Theorem 2 and Lemma 5, we can prove the following theorem.

Theorem 8. * The minimal polynomial of the level- scaled factor circulant matrix is the monic polynomial that generates the ideal
**
where the polynomial is the representer of .*

* Proof. *Consider the following -algebra homomorphism:
It is clear that if and only if . In view of Lemma 5, we have

We know from Theorem 8 and Lemma 6 that the minimal polynomial and the inverse of a level- scaled factor circulant matrix is calculated by a Gröbner basis for a kernel of an -algebra homomorphism. Therefore, we have the following algorithm to calculate the minimal polynomial and the inverse of a level- scaled factor circulant matrix .

*Step 1. *Calculate the reduced Gröbner basis for the ideal
by CoCoA 4.0, using an elimination order with .

*Step 2. *Find the polynomial in in which the variables do not appear. This polynomial is the minimal polynomial of .

*Step 3. *By Step 2, if in the minimal polynomial of ,
is zero; stop. Otherwise, calculate

*Example 9. *Let be a level-2 scaled factor circulant matrix, where
We now calculate the minimal polynomial and the inverse of with entries in field .

In fact, the reduced Gröbner basis for the ideal is So, the minimal polynomial of is and the inverse of is

Theorem 10. * The annihilation ideal of the level- scaled factor circulant matrices is
**
where the polynomial is the representer of .*

* Proof. *Consider the following -algebra homomorphism:
It is clear that if and only if . Hence, by Lemma 5

According to Theorem 10, we give the following algorithm for the annihilation ideal of the level- scaled factor circulant matrices .

*Step 4 . *Calculate the reduced Gröbner basis for the ideal
by CoCoA 4.0, using an elimination order with .

*Step 5. *Find the polynomial in in which the variables do not appear. Then, the ideal generated by these polynomials is the annihilation ideal of .

*Example 11. * Let and be both level-2 scaled circulant factor matrices, where
We calculate the annihilation ideal of and over field as follows.

By CoCoA 4.0, we obtain that the reduced Gröbner basis for the ideal is So, the annihilation ideal of and is

To calculate the common minimal polynomial of , we first prove the following theorem.

Theorem 12. * Let be the least common multiple of . Then,
*

*Proof. *For any , we have for . Since is the least common multiple of . So . Hence

Conversely, for because is the least common multiple of . Therefore,

Let be level- scaled factor circulant matrix for . If the minimal polynomial of is for , then the common minimal polynomial of is the least common multiple of . By Theorem 12 and Lemma 7, we have the following algorithm for finding the common minimal polynomial of level- scaled factor circulant matrices for .

*Step 6 . *Calculate the Gröbner basis for the ideal by CoCoA 4.0 for each , using an elimination order with .

*Step 7. *Find out the polynomial in in which the variables do not appear for each .

*Step 8. *Calculate the Gröbner basis for the ideal
by CoCoA 4.0, using elimination with .

*Step 9. *Find out the polynomial in in which the variables do not appear. Then, the polynomial is the common minimal polynomial of for .

*Example 13. *We now calculate the common minimal polynomial of and of Example 11 over field as follows.

By CoCoA 4.0, we obtain that the reduced Gröbner basis for the ideal is So, the minimal polynomial of is

Similarly, we get that the reduced Gröbner basis for the ideal is Thus, the minimal polynomial of is In addition, we obtain that the reduced Gröbner basis for the ideal is So, the common minimal polynomial of and is

#### 4. Inverses of Level- Scaled Factor Circulant Matrices

In this section, we discuss the nonsingularity and the inverse of a level- scaled factor circulant matrix.

Theorem 14. * Let be an level- scaled factor circulant matrix. Then, is nonsingular if and only if
**
where the polynomial is the representer of .*

* Proof. * is nonsingular if and only if
is an invertible element in
By Theorem 2, if and only if there exists such that if and only if there exist such that
if and only if

Let be an level- scaled factor circulant matrix. By Theorem 14, we have the following algorithm which can find the inverse of the matrix .

*Step 10 . *Calculate the reduced Gröbner basis for the ideal
where the polynomial is the representer of , by CoCoA 4.0, using a given term order with . If , then is singular. Stop. Otherwise, go to Step 11.

*Step 11. *Using Buchberger’s algorithm for computing Gröbner bases, by keeping track of linear combinations that give rise to the new polynomials in the generating set, we get such that

*Step 12. *The variables in formula (57) are replaced by , respectively. We have

#### 5. Inverse of Partitioned Matrix with Level- Scaled Factor Circulant Matrix Blocks

Let , and be level- scaled factor circulant matrices with the representers , and , respectively. If is nonsingular, let Then, So, is nonsingular if and only if is nonsingular. Since are all level- scaled factor circulant matrices, then commutes with if . Thus, From (61), we conclude that is nonsingular if and only if is nonsingular. Since is the representer of , then is nonsingular if and only if Furthermore, if is nonsingular, by (60), we have where .

We summarize our discussion as the following.

Theorem 15. *Let
**
where , and are all level- scaled factor circulant matrices with the representers , and , respectively. If is nonsingular, then is nonsingular if and only if
**
Moreover, if is nonsingular, then
**
where .*

Theorem 16. *Let
**
where , and are all level- scaled factor circulant matrices with the representers , and , respectively. If is nonsingular, then is nonsingular if and only if
**
In addition, if is nonsingular, then
**
where .*

*Proof. *Since is nonsingular, then
So. is nonsingular if and only if is nonsingular. Since , and are all level- scaled factor circulant matrices, then commutes with if . Thus,
By (71), we conclude that is nonsingular if and only if is nonsingular. Since is the representer of , then is nonsingular if and only if
If is nonsingular, by (70), we have
where .

We have the following algorithm for determining the nonsingularity and computing the inverse of if it is nonsingular.

*Step 13 . *Calculate the bases for the ideals
respectively. If Stop. Otherwise, go to Step 14.

*Step 14. *If , find such that
Then, is the representer of , and go to Step 16. Otherwise, go to Step 15.

*Step 15. *If , find such that
Then, is the representer of , and go to Step 16.

*Step 16. *Calculate the Gröbner bases for the ideal
If , then is singular, Stop. Otherwise, go to Step 17.

*Step 17. *Find such that
Then, is the representer of . Thus, we obtain that

if is nonsingular, then
where .

If is nonsingular, then
where .

#### 6. Conclusion

Algorithms for finding the minimal polynomial of the level- scaled factor circulant matrices over any field are presented. And two algorithms for finding the inverses of such matrices are also presented. Finally, an algorithm for computing the inverse of partitioned matrix with level- scaled factor circulant matrix blocks over any field is given. In the future, we will investigate the application in solving various differential equations based on multilevel circulant matrices.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The research was supported by the Development Project of Science & Technology of Shandong Province (Grant no. 2012GGX10115) and the AMEP of Linyi University, China.

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