Advances in Linear Algebra & Matrix Theory
Vol.05 No.02(2015), Article ID:57509,7 pages
10.4236/alamt.2015.52006

The Discriminance for FLDcircr Matrices and the Fast Algorithm of Their Inverse and Generalized Inverse

Xue Pan, Mei Qin

College of Science, University of Shanghai for Science and Technology, Shanghai, China

Email: xiaoxuehe1126@163.com   Received 26 May 2015; accepted 23 June 2015; published 29 June 2015

ABSTRACT

This paper presents a new type of circulant matrices. We call it the first and the last difference r-circulant matrix (FLDcircr matrix). We can verify that the linear operation, the matrix product and the inverse matrix of this type of matrices are still FLDcircr matrices. By constructing the basic FLDcircr matrix, we give the discriminance for FLDcircr matrices and the fast algorithm of the inverse and generalized inverse of the FLDcircr matrices.

Keywords:

FLDcircr Matrix, Discriminance, Diagonalization, Inverse, Generalized Inverse 1. Introduction

Circulant matrix plays an important role in the matrix theory, its special structure and properties have been widely used in applied mathematics, physics, modern engineering, and so on  - . There have been many new circulant matrices come fordward  - . In this paper we will firstly put forward the concept of the FLDcircr matrix and the basic FLDcircr matrix. The sum, the difference, the product, the inverse and the adjoint matrix of this type of matrices are still FLDcircr matrices. Then, we will give five discriminance for FLDcircr matrix by constructing the basic FLDcircr matrix. At last, we will discuss the fast algorithm of the inverse and generalized inverse of the FLDcircr matrix and give the numerical example. In this paper, we just study the square matrices in complex field.

2. Definition of the FLDcircr Matrix

Definition 2.1 For a square matrix A of order n, if its form is ,

We call it the FLDcircr matrix, and denote shortly .

Definition 2.2 Let D is the basic FLDcircr matrix of order n, that is .

We obtain is the characteristic polynomial of D, , we specify .

From the definition of FLDcircr matrix, we can prove the following proposition.

Proposition 2.3 If A and B are FLDcircr matrices, then A + B, A − B and kA are both FLDcircr matrices, for any k belongs to the complex field.

Definition 2.4 Let ,the index of A is the least nonnegative integer k such that , we note it as . If A is nonsingular, then ; if A is singular, then .

Definition 2.5 Let , if there is which satisfies , at the same time, we named X as the reflexive generalize inverse of A, we note it as .

Definition 2.6 Let, , if satisfies

Then we denote X as the Drazin inverse of A, note it as.

Lemma 2.7 If polynomial matrix can transformed into after elemen-

tary row transformation, then we have, and.

3. The Discriminance of the FLDcircr Matrix

Theorem 3.1 A is an FLDcircr matrix if and only if A is of the following form

(1)

For some polynomial.

Proof. By the Definition 2.1 and Definition 2.2, we get this result.

Theorem 3.2 A is an FLDcircr matrix if and only if AD = DA, D is the basic FLDcircr matrix.

Proof. For A is an FLDcircr matrix, from the definition of A and D, we obtain

By the method of undetermined coefficients, let

.

Due to

It follows that

We obtain

,

So A is an FLDcircr matrix.

Corollary 3.3 If A and B are both FLDcircr matrices, then AB and BA are FLDcircr matrices. Furthermore, we get AB = BA.

Proof. Since A and B are FLDcircr matrices, by the Theorem 3.2, we get

Hence

Then, AB and BA are both FLDcircr matrices.

From Theorem 3.1, we have

4. The Diagonalization of the FLDcircr Matrix

First, we consider the diagonalization of the basic FLDcircr matrix D.

For the characteristic polynomial of D has n different roots. So, D has n different eigenvalues:

.

Let

,

Obviously, is a nonsingular Vandermonde matrix about, and

. (2)

Next, we study the diagonalization of general FLDcircr matrix A.

From Theorem 3.1 and Equation (2), we obtain

The eigenvalues of A are

.

Theorem 4.1 A is an FLDcircr matrix if and only if is a diagonal matrix.

Proof. If A is an FLDcircr matrix, from the above discussion, we have

.

Let, is a diagonal matrix, then

.

Let, from Equation (2) we have

,

Thus

For and are both diagonal matrix, so

hence, A is an FLDcircr matrix.

Theorem 4.2 A is a nonsingular FLDcircr matrix if and only if the eigenvalues, where are eigenvalues of the basic FLDcircr matrix.

Proof. For A is a nonsingular FLDcircr matrix, from the above discussion, we have

,

where are eigenvalues of A.

So

.

Hence, if A is a nonsingular FLDcircr matrix, we have.

Due to,

Then

,

So A is nonsingular.

5. The Fast Algorithm of the Inverse and Generalized Inverse of the FLDcircr Matrix

Theorem 5.1 If A is a nonsingular matrix, then A is an FLDcircr matrix if and only if is an FLDcircr matrix.

Proof. From A is nonsingular and Theorem 3.2, we obtain

Hence

That is to say is an FLDcircr matrix.

Clearly, the nonsingular matrix A is an FLDcircr matrix.

Corollary 5.2 If A is a nonsingular FLDcircr matrix, then is a nonsingular FLDcircr matrix.

Proof. For A is an FLDcircr matrix, we have, so

.

Due to

,

Thus

,

Hence

Then is an FLDcircr matrix.

Theorem 5.3 If A is an FLDcircr matrix, then A is nonsingular if and only if.

Proof. If A is a nonsingular FLDcircr matrix, from Theorem 4.2, we have, so and don’t have the same solutions, thus.

Otherwise, if, there exist, such that, . For, , we have. So, A is nonsingular and. From Theorem 3.1, we have is an FLDcircr matrix.

Corollary 5.4 If A is a nonsingular FLDcircr matrix, there exits.

Corollary 5.5 A is a singular FLDcircr matrix, there exists an FLDcircr matrix H that satisfies .

Proof. For A is singular, we get. Suppose, ,

, then. Furthermore, doesn’t have repeated root, thus,

, ,. So,.

Hence, there exist, such that

. (3)

Equation (3) both sides multiplied by, then

.

For, , we have

(4)

Equation (3) both sides multiplied by. Similarly, we get

. (5)

If, then H is the polynomial of D, from Theorem 3.1, we get H is an FLDcircr matrix, and from Equation (4), Equation (5) we have.

Due to

Hence.

From Lemma 2.7 and the proof of Theorem 5.3, Corollary 5.5, we can get the fast algorithm of the inverse and generalized inverse of the FLDcircr matrix. The general steps are as follows:

Step 1 get the greatest common factor of,;

Step 2 If, the polynomial matrix can transformed into after elementary

row transformation, then;

Step 3 If, divide by, get, then the polynomial matrix can transformed into after elementary row transformation, hence.

Example 5.1 If the 3 order matrix, then whether A is a nonsingular matrix? If A is non-

singular, solving.

From Definition 2.1 we get, , ,. Because of, so A is nonsingular.

After a series of elementary row transformation of the following polynomial matrix, we obtain

So

.

Therefore

,

That is

.

Example 5.2 If the 3 order matrix, solving.

From Definition 2.1 we have, , ,.

Then, so, A is singular and.

From Step 3, we get

Then

So

,

That is

.

Acknowledgements

The authors are grateful to the anonymous referees for their review comments and suggestions that help to improve the original manuscript.

References

1. Searle, S.R. (1979) On Inverting Circulant Matrices. Linear Algebra and Its Applications, 25, 77-89. http://dx.doi.org/10.1016/0024-3795(79)90007-7
2. Tolomei, P.B. and Torres, L.M. (2013) On the First Chvátal Closure of the Set Covering Polyhedron Related to Circulant Matrices. Electronic Notes in Discrete Mathematics, 44, 377-383. http://dx.doi.org/10.1016/j.endm.2013.10.059
3. Bozkurt, D. and Tam, T.-Y. (2012) Determinants and Inverses of Circulant Matrices with Jacobsthal and Jacobsthal-Lucas Numbers. Applied Mathematics and Computation, 219, 544-551. http://dx.doi.org/10.1016/j.amc.2012.06.039
4. Wu, A.-G., Zeng, X.L., Duan, G.-R. and Wu, W.-J. (2010) Iterative Solutions to the Extended Sylvester-Conjugate Matrix Equations. Applied Mathematics and Computation, 217, 130-142. http://dx.doi.org/10.1016/j.amc.2010.05.029
5. Shen, S.Q. and Cen, J.M. (2010) On the Bounds for the Norms of r-Circulant Matrices with the Fibonacci and Lucas Numbers. Applied Mathematics and Computation, 216, 2891-2897. http://dx.doi.org/10.1016/j.amc.2010.03.140
6. Zellini, P. (1979) On Some Properties of Circulant Matrices. Linear Algebra and Its Applications, 26, 31-43. http://dx.doi.org/10.1016/0024-3795(79)90170-8
7. Davis, P.J. (1994) Circulant Matrices. 2nd Edition, Chelsea Publishing, New York.
8. Xu, Q.-Z., Li, H.-Q. and Jiang, Z.-L. (2005) An Algorithm for Finding the Minimal Polynomial of Fls r-Circulant Matrix. Journal of Mathematics (PRC), 25, 599-604.
9. He, C.Y. and Wang, X.Y. (2014) The Discriminance for a Special Class of Circulant Matrices and Their Diagonalization. Applied Mathematics and Computation, 238, 7-12. http://dx.doi.org/10.1016/j.amc.2014.04.007
10. Zhou, J.W. and Jiang, Z.L. (2014) The Spectral Norms of g-Circulant Matrices with Classical Fibonacci and Lucas Numbers Entries. Applied Mathematics and Computation, 233, 582-587. http://dx.doi.org/10.1016/j.amc.2014.02.020
11. Bell, C.L. (1981) Generalized Inverses of Circulant and Generalized Circulant Matrices. Linear Algebra and Its Applications, 39, 133-142. http://dx.doi.org/10.1016/0024-3795(81)90297-4
12. Jiang, Z.-J. and Xu, Z.-B. (2005) Efficient Algorithm for Finding the Inverse and the Group Inverse of FLSr-Circulant Matrix. Journal of Applied Mathematics and Computing, 18, 45-57. http://dx.doi.org/10.1007/BF02936555