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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
FLDcirc_{r} matrices. By constructing the basic
FLDcirc_{r} matrix, we give the discriminance for
FLDcirc_{r} matrices and the fast algorithm of the inverse and generalized inverse of the
FLDcirc_{r} matrices.

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 [_{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

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

Definition 2.5 Let

Definition 2.6 Let

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

Lemma 2.7 If polynomial matrix

tary row transformation, then we have

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. _{r} matrix, from the definition of A and D, we obtain

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

Let

Obviously,

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

Proof. _{r} matrix, from the above discussion, we have

Let

Thus

For

hence, A is an FLDcirc_{r} matrix.

Theorem 4.2 A is a nonsingular FLDcirc_{r} matrix if and only if the eigenvalues_{r} matrix.

Proof. _{r} matrix, from the above discussion, we have

where

So

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

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 _{r} matrix.

Proof.

Hence

That is to say _{r} matrix.

_{r} matrix.

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

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

Due to

Thus

Hence

Then _{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

Otherwise, if_{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

Hence, there exist

Equation (3) both sides multiplied by

For

Equation (3) both sides multiplied by

If_{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

Step 2 If

row transformation, then

Step 3 If

Example 5.1 If the 3 order matrix

singular, solving

From Definition 2.1 we get

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

From Definition 2.1 we have

Then

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.