Displacement structure of the DMP inverse

: A matrix A is said to have the displacement structure if the rank of the Sylvester displacement AU VA − or the Stein displacement A VAU − is much smaller than the rank of A . In this article, we study the displacement structure of the DMP inverse A d ,† . Estimations for the Sylvester displacement rank of DMP inverse are presented under some restrictions. The generalized displacement is also discussed. The general results are applied to the core inverse.


Introduction
Let m n × be the set of all m n × complex matrices. For A m n ∈ × , A ( ), A Ker( ), A * , and A T denote the range, kernel, conjugate transpose, and transpose of A, respectively. I n is the identity matrix of order n. The index of A n n ∈ × , denoted by A Ind( ), is the smallest nonnegative integer k such that A A rank rank k k1 The present article is to study the displacement structure of the DMP inverse [1], which is motivated by [2][3][4][5][6][7][8][9][10].
To begin with, we shall recall definitions of some generalized inverses. The Moore-Penrose inverse of a matrix A m n ∈ × , denoted by A † , is the unique matrix X n m ∈ × satisfying the following Penrose equations [11]: where k is the index of A. When A Ind 1 ( ) = , A d is called the group inverse of A and is denoted by A # , and see [11].
In [12], Baksalary and Trenkler introduced the notion of core inverse for a square matrix of index 1. Let A n n ∈ × be with A Ind 1 ( ) = . Then the core inverse of A is defined as the unique matrix X n n ∈ × satisfying and is denoted by X A = ○ # . It is known that A A AA † = ○ # # . Then three generalizations of the core inverse were recently introduced for n n × complex matrices, namely, core-EP inverse [13], BT inverse [14], and DMP inverse [15]. The DMP inverse [16] [17][18][19][20]. Furthermore, the definition of the DMP inverse of a square matrix was extended to rectangular matrices in [21], which is called the W -weighted DMP inverse.
Malik and Thome [15] gave the canonical form for the DMP inverse of a square matrix by using the Hartwig-Spindelböck decomposition (see [22]), which is useful in analyzing the properties of the DMP inverse. For any A n n ∈ × of rank r 0 > , the Hartwig-Spindelböck decomposition is given by is a diagonal matrix, and the diagonal entries The concept of the displacement structure was presented in [23] for the inverse of an integral operator with a convolution kernel. A matrix A is said to have the displacement structure if we can find twodimensionally compatible matrices U and V such that the rank of the Sylvester displacement AU VA − or the Stein displacement A VAU − is much smaller than the rank of A [24]. It is well known that fast inversion algorithms for a matrix A can be constructed if A is a matrix with a displacement structure. The displacement structure is commonly exploited in the computation of generalized inverses. In recent years, displacement structures of various generalized inverses, such as Moore-Penrose inverse, weighted Moore-Penrose inverse, group inverse, M-group inverse, Drazin inverse, W -weighted Drazin inverse, and core inverse, were studied, and see [2][3][4][5][6][8][9][10].
In this article, we will give an upper bound for the Sylvester displacement rank of the DMP inverse under some restrictions in Section 2. In Section 3, we will consider a more general displacement. An estimation for the generalized displacement rank of the DMP inverse under some restrictions is presented. The general results are applied to the core inverse.

Sylvester displacement rank
) is called the Sylvester displacement rank of A. It is well known that the Sylvester displacement rank of a Toeplitz rank is at most 2. This low displacement rank property can be exploited to develop fast algorithms for triangular factorization, and inversion, among others [24]. For a nonsingular matrix A, the equality AU VA A UA tells us that the Sylvester displacement rank of a nonsingular matrix A equals that of its inverse A 1 − . In other words, if A is structured with respect to the Sylvester displacement rank associated with U V , ( ), then its inverse A 1 − is also structured with respect to the Sylvester displacement rank associated with V U , ( ). It is natural to consider the displacement rank of generalized inverses. In this section, we will give an upper bound for the displacement rank of DMP inverse under some restrictions.
Then it is easy to see that A AU VA A I N UA = − − − * Thus, we obtain the following lemma.
By considering the ranks of both sides in equality (3), we can obtain an upper bound for the Sylvester displacement rank of A d, † .
Theorem 2.1. The VU -displacement rank of A d, † satisfies the following estimate: Next we give an upper bound for the sum of the second and third terms on the right-hand side of (4) under some restrictions.
Displacement structure of the DMP inverse  1205 Proof. We first determine the structure of G. If A has the form (1), then it is not difficult to see that Then On the other hand, We can see from (8) Notice that It follows from [25], (7), and (10) that Then it can be seen from K rank Σ 3 ( ) = and U V rank rank 4 11 11 ( ) ( ) A direct calculation shows that Let A n n ∈ × be of the form (1).
We remark that the author in [6] showed that (11) holds for G AA = * , while we derive a different G such that the inequality (11) holds.

Generalized displacement rank
In this section, we will consider a more general displacement, and the results of Sylvester displacement in Section 2 will be extended.
Let a a ij 0 be a nonsingular 2 2 × matrix. For any fixed U n n If we set a a 0 1 1 0 and respectively, in (12) We first give two lemmas, which are taken from [4].
where G AA † = . In this section, we study the explicit form of the displacement, and we give an expression for the displacement of the DMP inverse of A through DMP inverse solutions of some special linear systems of equations. For simplicity, we only consider the Sylvester displacement, and the starting point is (3