Research Article | Open Access
Xiaoji Liu, Fang Gui, "Some Characterizations and Properties of a New Partial Order", Journal of Mathematics, vol. 2020, Article ID 3215038, 9 pages, 2020. https://doi.org/10.1155/2020/3215038
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.
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  introduced generalized Löwner order by polar decomposition; Baksalary and Trenkler  studied the core partial order of complex matrices; and Ando  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.
Lemma 1 (see ). Let with , then can be expressed aswhere 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 (, Chapter 6, Theorem 7). Let . Then, can be written aswhere 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  introduced the GL partial order: let , and and be their polar decompositions, where and . Then,
After that, Wang and Liu  made the polar-like decomposition: let . Then, can be written aswhere , , and are given in (, Chapter 6, Theorem 7). On the basis of the polar-like decomposition, the WL partial order  is defined asin which and are the polar-like decompositions of and , respectively. In , 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 thatwhere , and are both nonsingular, , , , and are arbitrary matrices with appropriate sizes, ,
Proof. Let . Applying (11), we get , whereFurthermore, we writewhere and .
Let , and be the P-2 expression of . Since , by applying (14), we have . Then, by applying (, Lemma 3), we getwhere is nonsingular. It is easy to check thatWe writeThen, applying (13), we get , andSince and , then we get , andthat 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. LetThen, , , andSincethen . 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. Letin whichThen, , , andSinceso . 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 haveThen,Therefore, .
Corollary 1 (see ). Let . Then, if and only if .
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 thatwhere , , and are both nonsingular, and .
Proof. Let be EP, , . There exists a unitary matrix such thatThen, , . Furthermore,Since is EP,and , we getThen,Since , , and and are both EP, we have , and . And because and , thenTherefore,So, we have (41).
It is noteworthy that , when and both are EP.
Example 3. Let, andWe obtain . SinceWe have . But is not below under the Löwner partial order. So, .
All data generated or analysed during this study are included in this published article.
Conflicts of Interest
The authors declare no conflicts of interest.
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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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.