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.

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.

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

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

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

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.

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

Equation (3) both sides multiplied by, then

.

For, , we have

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

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

.

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