Some Characterizations and Properties of a New Partial Order

On the basis of Löwner partial order and core partial order, we introduce a new partial order: LC partial order. By applying matrix decomposition, core inverse, core partial order, and Löwner partial order, we give some characteristics of LC partial order, study the relationship between LC partial order and Löwner partial order under constraint conditions, and illustrate its differences with some classical partial orders, such as minus, CL, and GL partial orders.


Introduction
A binary relation on a nonempty set is called partial order if it satisfies reflexivity, transitivity, and antisymmetry. In recent years, more and more mathematicians have turned their attention to matrix partial ordering: Hauke and Markiewicz [1] introduced generalized Löwner order by polar decomposition; Baksalary and Trenkler [2] studied the core partial order of complex matrices; and Ando [3] studied the square inequality and strong order relation on Hilbert space. In this paper, a new partial order is introduced on the complex matrix set by matrix decomposition and Löwner and core partial orders.
First, we use the following notations. e symbol C m,n denotes the set of m × n matrices with complex entries. C H n and C ≥ n denote the set of n × n Hermitian matrices and Hermitian nonnegative definite matrices, respectively. e symbols A * , R(A), and rk(A) represent the conjugate transpose, range space, and rank of A ∈ C m,n , respectively. e symbol ρ(A) represents spectral radius of A ∈ C n,n . e smallest positive integer k for which rk(A k+1 ) � rk(A k ) is called the index of A ∈ C n,n and is denoted by Ind(A). When A is nonsingular, the index of A is 0. e symbol C CM n stands for a set of n × n matrices of index less than or equal to 1.
Definition 1 (see [4,5]). Let A ∈ C m,n . If X ∈ C n,m satisfies the following equations: then X is said to be the Moore-Penrose inverse of matrix A, and X is unique. It is usually defined by X � A † .
Furthermore, we denote P A � AA † .
Definition 2 (see [4,5]). Let A ∈ C CM n . If X ∈ C n,n satisfies the following equations: then X is said to be the core inverse of matrix A, and X is unique. It is usually defined by X � A # .
Matrix decomposition is an important tool to study the theory of matrix partial orders. It is used to discuss some characteristics and properties of matrix partial orders and then to establish some matrix partial orders. For example, C-N partial order and core partial order are based on C-N decomposition and core decomposition, respectively [2,7,9].
A particular concern is the generalized polar decomposition ( [3], Chapter 6, eorem 7). Let A ∈ C m,n . en, A can be written as where E A ∈ C m,n is a partial isometry, i.e., E * A � E † A , G A ∈ C m,m , and H A ∈ C n,n are Hermitian nonnegative definite matrices. e matrices E A , G A , and H A are uniquely determined by Based on the generalized polar decomposition, Hauke and Markiewicz [1] introduced the GL partial order: let After that, Wang and Liu [10] made the polar-like decomposition: let A ∈ C m,n . en, A can be written as where E A , G A , and H A are given in ( [3], Chapter 6, eorem 7). On the basis of the polar-like decomposition, the WL partial order [10] is defined as B are the polarlike decompositions of A and B, respectively. In [11], Wang and Liu introduced the CL partial order: in which A, B ∈ C CM n . It is also worthy to note that, under certain conditions, CL partial order is equivalent to GL and Löwner partial orders [1,11].
In this paper, we consider matrices over complex fields. Based on the above research and inspired by generalized polar decomposition, we introduce a new partial order on the set of core matrices by using Löwner partial order and core partial order. It is dominated neither by minus partial order nor by Löwner partial order. Interestingly, under some conditions, LC partial order is equivalent to CL, GL, and Löwner partial orders.

Main Result
In this section, we introduce a new partial order on C CM n , derive some of its characteristics, consider its relationship with Löwner partial order under some constraints, and illustrate its difference from other partial orders with examples.
Let A ∈ C CM n , P A � AA † , and E A be as given in (5). en, where We call (10) the P-2 expression of A. Furthermore, it is easy to check that F A ∈ C n,n is a EP-matrix, ρ(F A ) ≤ 1, and Q A ∈ C n,n . Let A, B ∈ C CM n . Consider the binary operation: in which A � F A Q A and B � F B Q B are the P-2 expressions of A and B, respectively. In the following eorem 1, we check that the binary operation is a partial order and call it the LC partial order. (14) is a partial order on C CM n .

Proof
(1) Reflexive: let A ∈ C CM n , and A � F A Q A is the P-2 expression of A. We have From the transitivity of the Löwner partial order and core partial order, we have F A ≤ L F C and Q A ≤ ○ # Q C , that is, A ≤ LC C. By (1), (2), and (3), we know that the binary operation (14) is a partial order on C CM n .
Next, we give the characteristics of the LC partial order. □ where A 1 , B 1 ∈ C rk(A),rk(A) , T A1 and T A4 are both nonsingular, T A2 , T A3 , T A4 , and T A5 are arbitrary matrices with appropriate sizes, ρ( Furthermore, we write where where T A4 is nonsingular. It is easy to check that We write en, applying (13), we get B 7 � 0, B 8 � 0, B 9 � 0, and

Journal of Mathematics
Since that is, Next, we use Examples 1 and 2 to explain the difference between LC partial order and minus (Löwner, GL, or CL) partial order.
en, rk(B) � 3, rk(A) � 2, and Since 4 Journal of Mathematics is not below B under the Löwner partial order in which

Journal of Mathematics
Since (1) But is not a positive semidefinite matrix, so A is not below B under the CL partial order.
(2) Since A is not below B under the GL partial order.
Proof. When rk(A) � rk(B), we have en, □ Corollary 1 (see [7]). (40) An EP matrix is core invertible, and the core, Moore-Penrose, and group inverses of the matrix are identical. Next, we consider the case where both A and B are EP. en, there exists a unitary matrix U such that Journal of Mathematics and T A4 are both nonsingular, and en, L A � 0, K A � I rk(A) . Furthermore, Since B is EP, and rk(B) ≥ rk(A) ≥ 1, we get en, Since ρ(K A ) ≤ 1, ρ(K B ) ≤ 1, and F A and F B are both EP, we have erefore,  (50) We obtain Q B ≤ ○ # Q A . Since

Data Availability
All data generated or analysed during this study are included in this published article.

Conflicts of Interest
e authors declare no conflicts of interest.