Journal of Mathematics
Volume 2020 (2020), Article ID 3215038, 9 pages
https://doi.org/10.1155/2020/3215038

Research Article

Some Characterizations and Properties of a New Partial Order

Xiaoji Liu and Fang Gui

School of Mathematics and Physics, Guangxi University for Nationalities, Nanning 530006, China

Correspondence should be addressed to Xiaoji Liu; xiaojiliu72@126.com

Received 11 June 2020; Accepted 11 August 2020; Published 9 September 2020

Academic Editor: Jia-Bao Liu

Copyright © 2020 Xiaoji Liu and Fang Gui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. 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. The symbol denotes the set of matrices with complex entries. and denote the set of Hermitian matrices and Hermitian nonnegative definite matrices, respectively. The symbols , , and represent the conjugate transpose, range space, and rank of , respectively. The symbol represents spectral radius of .

The smallest positive integer for which is called the index of and is denoted by . When is nonsingular, the index of is 0. The symbol stands for a set of matrices of index less than or equal to 1.

Definition 1 (see [4, 5]). Let . If satisfies the following equations:

then is said to be the Moore–Penrose inverse of matrix , and is unique. It is usually defined by .
Furthermore, we denote .

Definition 2 (see [4, 5]). Let . If satisfies the following equations:

then is said to be the core inverse of matrix , and is unique. It is usually defined by .

Lemma 1 (see [2]). Let with , then can be expressed as

where is unitary, is the diagonal matrix of singular values of , , and satisfy , and .

Furthermore, when , is nonsingular, and

We give the definitions of some classical partial orders such as minus, Löwner, sharp, core, and C-N partial orders [2, 6–8].

(1)(2)(3)(4)(5), and , in which and are the C-N decompositions of and , respectively

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, Theorem 7). Let . Then, can be written as

where is a partial isometry, i.e., , , and are Hermitian nonnegative definite matrices. The matrices , , and are uniquely determined by and , in which , , and .

Based on the generalized polar decomposition, Hauke and Markiewicz [1] introduced the GL partial order: let , and and be their polar decompositions, where and . Then,

After that, Wang and Liu [10] made the polar-like decomposition: let . Then, can be written as

where , , and are given in ([3], Chapter 6, Theorem 7). On the basis of the polar-like decomposition, the WL partial order [10] is defined as

in which and are the polar-like decompositions of and , respectively. In [11], Wang and Liu introduced the CL partial order:

in which . 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.

2. Main Result

In this section, we introduce a new partial order on , 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 , , and be as given in (5). Then,

where

We call (10) the P-2 expression of . Furthermore, it is easy to check that

is a EP-matrix, , and .

Let . Consider the binary operation:

in which and are the P-2 expressions of and , respectively. In the following Theorem 1, we check that the binary operation is a partial order and call it the LC partial order.

Theorem 1. The binary operation (14) is a partial order on .

Proof.

(1)Reflexive: let , and is the P-2 expression of . We have

 So, .(2)Antisymmetric: let , and be their P-2 expressions. If and , that is, if

 Then, .(3)Transitive: suppose that and , that is, suppose

 From the transitivity of the Löwner partial order and core partial order, we have and , that is, . By (1), (2), and (3), we know that the binary operation (14) is a partial order on .

Next, we give the characteristics of the LC partial order.

Theorem 2. Let , , , and , and be the P-2 expressions of and , respectively. Then, there exists a unitary matrix such that

where , and are both nonsingular, , , , and are arbitrary matrices with appropriate sizes, ,

Proof. Let . Applying (11), we get , where

Furthermore, we write

where and .
Let , and be the P-2 expression of . Since , by applying (14), we have . Then, by applying ([2], Lemma 3), we get

where is nonsingular. It is easy to check that

We write

Then, applying (13), we get , and

Since and , then we get , and

that is,

Since , , and are EP, and , we obtain and . Therefore, we have (18).
Next, we use Examples 1 and 2 to explain the difference between LC partial order and minus (Löwner, GL, or CL) partial order.

Example 1. Let

Then, , , and

Since

then . We have .

(1)Since , is not below under the minus partial order(2)Since is not a positive semi-definite matrix, is not below under the Löwner partial order

Example 2. Let

in which

Then, , , and

Since

so . We have .
(1) But

is not a positive semidefinite matrix, so is not below under the CL partial order.
(2) Since

is not below under the GL partial order.

Theorem 3. Let , . Then,

Proof. When , we have

Then,

Therefore, .

Corollary 1 (see [7]). Let . Then, if and only if .

In ([8], Theorem 8.5.15), Mitra has verified that when , there has if and only if . By applying Theorem 3, Corollary 1, and ([8], Theorem 8.5.15), we get Corollary 2.

Corollary 2. Let and . Then,

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 and are EP.

Theorem 4. Let be EP, , , , and be the P-2 expression of and . Then, there exists a unitary matrix such that

where , , and are both nonsingular, and .

Proof. Let be EP, , . There exists a unitary matrix such that

Then, , . Furthermore,

Since is EP,

and , we get

Then,

Since , , and and are both EP, we have , and . And because and , then

Therefore,

So, we have (41).
It is noteworthy that , when and both are EP.

Example 3. Let

, and

We obtain . Since

We have . But

is not below under the Löwner partial order. So, .

Data Availability

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

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

The work was supported by Guangxi Natural Science Foundation (No. 2018GXNSFDA281023), High Level Innovation Teams and Distinguished Scholars in Guangxi Universities (No. GUIJIAOREN201642HAO), the National Natural Science Foundation of China (No. 11361009), the Special Fund for Bagui Scholars of Guangxi (No. 2016A17), and the Education Innovation Program for 2019 Graduate Students (No. gxun-chxzs2019026).

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