The Discriminance for *FLDcirc*_{r} 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.

**DOI: **10.4236/alamt.2015.52006
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College of Science, University of Shanghai for Science and Technology, Shanghai, China.

This
paper presents a new type of circulant matrices. We call it the first and the
last difference *r*-circulant matrix (*FLDcirc _{r}* matrix). We can
verify that the linear operation, the matrix product and the inverse matrix of
this type of matrices are still

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Pan, X. and Qin, M. (2015) The Discriminance for *FLDcirc*_{r} Matrices and the Fast Algorithm of Their Inverse and Generalized Inverse. *Advances in Linear Algebra & Matrix Theory*, **5**, 54-61. doi: 10.4236/alamt.2015.52006.

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 [1] -[6] . There have been many new circulant matrices come fordward [7] -[12] . In this paper we will firstly put forward the concept of the FLDcirc_{r} matrix and the basic FLDcirc_{r} matrix. The sum, the difference, the product, the inverse and the adjoint matrix of this type of matrices are still FLDcirc_{r} matrices. Then, we will give five discriminance for FLDcirc_{r} matrix by constructing the basic FLDcirc_{r} matrix. At last, we will discuss the fast algorithm of the inverse and generalized inverse of the FLDcirc_{r} matrix and give the numerical example. In this paper, we just study the square matrices in complex field.

2. Definition of the FLDcirc_{r} Matrix

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

,

We call it the FLDcirc_{r} matrix, and denote shortly.

Definition 2.2 Let D is the basic FLDcirc_{r} matrix of order n, that is

.

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

From the definition of FLDcirc_{r} matrix, we can prove the following proposition.

Proposition 2.3 If A and B are FLDcirc_{r} matrices, then A + B, A − B and kA are both FLDcirc_{r} 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 FLDcirc_{r} Matrix

Theorem 3.1 A is an FLDcirc_{r} 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 FLDcirc_{r} matrix if and only if AD = DA, D is the basic FLDcirc_{r} matrix.

Proof. For A is an FLDcirc_{r} 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 FLDcirc_{r} matrix.

Corollary 3.3 If A and B are both FLDcirc_{r} matrices, then AB and BA are FLDcirc_{r} matrices. Furthermore, we get AB = BA.

Proof. Since A and B are FLDcirc_{r} matrices, by the Theorem 3.2, we get

Hence

Then, AB and BA are both FLDcirc_{r} matrices.

From Theorem 3.1, we have

4. The Diagonalization of the FLDcirc_{r} Matrix

First, we consider the diagonalization of the basic FLDcirc_{r} 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 FLDcirc_{r} matrix A.

From Theorem 3.1 and Equation (2), we obtain

The eigenvalues of A are

.

Theorem 4.1 A is an FLDcirc_{r} matrix if and only if is a diagonal matrix.

Proof. If A is an FLDcirc_{r} 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 FLDcirc_{r} matrix.

Theorem 4.2 A is a nonsingular FLDcirc_{r} matrix if and only if the eigenvalues, where are eigenvalues of the basic FLDcirc_{r} matrix.

Proof. For A is a nonsingular FLDcirc_{r} matrix, from the above discussion, we have

,

where are eigenvalues of A.

So

.

Hence, if A is a nonsingular FLDcirc_{r} matrix, we have.

Due to,

Then

,

So A is nonsingular.

5. The Fast Algorithm of the Inverse and Generalized Inverse of the FLDcirc_{r} Matrix

Theorem 5.1 If A is a nonsingular matrix, then A is an FLDcirc_{r} matrix if and only if is an FLDcirc_{r} matrix.

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

Hence

That is to say is an FLDcirc_{r} matrix.

Clearly, the nonsingular matrix A is an FLDcirc_{r} matrix.

Corollary 5.2 If A is a nonsingular FLDcirc_{r} matrix, then is a nonsingular FLDcirc_{r} matrix.

Proof. For A is an FLDcirc_{r} matrix, we have, so

.

Due to

,

Thus

,

Hence

Then is an FLDcirc_{r} matrix.

Theorem 5.3 If A is an FLDcirc_{r} matrix, then A is nonsingular if and only if.

Proof. If A is a nonsingular FLDcirc_{r} 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 FLDcirc_{r} matrix.

Corollary 5.4 If A is a nonsingular FLDcirc_{r} matrix, there exits.

Corollary 5.5 A is a singular FLDcirc_{r} matrix, there exists an FLDcirc_{r} 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 FLDcirc_{r} 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 FLDcirc_{r} 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.

Conflicts of Interest

The authors declare no conflicts of interest.

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