# Some Characterizations and Properties of a New Partial Order

Authors: Xiaoji Liu; Fang Gui
Journal: Journal of Mathematics
 (2020)

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## 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.


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## Body

## 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 symbolℂm,n denotes the set of m×n matrices with complex entries. ℂnH and ℂn≥ denote the set of n×n Hermitian matrices and Hermitian nonnegative definite matrices, respectively. The symbols A∗, ℛA, and rkA represent the conjugate transpose, range space, and rank of A∈ℂm,n, respectively. The symbol ρA represents spectral radius of A∈ℂn,n.The smallest positive integerk for which rkAk+1=rkAk is called the index of A∈ℂn,n and is denoted by IndA. When A is nonsingular, the index of A is 0. The symbol ℂnCM stands for a set of n×n matrices of index less than or equal to 1.Definition 1 (see [4, 5]).
LetA∈ℂm,n. If X∈ℂn,m satisfies the following equations:(1)1AXA=A,2XAX=X,3AX∗=AX,4XA∗=XA,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 denotePA=AA†.Definition 2 (see [4, 5]).
LetA∈ℂnCM. If X∈ℂn,n satisfies the following equations:(2)1AXA=A,2′AX2=X,3AX∗=AX,then X is said to be the core inverse of matrix A, and X is unique. It is usually defined by X=A#.Lemma 1 (see [2]).
LetA∈ℂn,n with rkA=r, then A can be expressed as(3)A=UΣKΣL00U∗,where U∈ℂn,n is unitary, Σ=diagα1,α2,…,αr is the diagonal matrix of singular values of A, α1≥α2≥⋯≥αr>0, K∈ℂr,r and L∈ℂr,n−r satisfy KK∗+LL∗=Ir, and ρK≤1.Furthermore, whenrkA=rkA2, K is nonsingular, and(4)A○#=UK−1Σ−1000U∗.We give the definitions of some classical partial orders such as minus, Löwner, sharp, core, and C-N partial orders [2, 6–8].(1)
A≤B,A,B∈ℂm,n,rkB−rkA=rkB−A(2)
A≤LB,A,B∈ℂn,n,B−A≥0(3)
A≤#B,A,B∈ℂnCM,A#A=A#BandAA#=BA#(4)
A≤○#B,A,B∈ℂnCM,A○#A=A○#BandAA○#=BA○#(5)
A≤#,−B:A,B∈ℂn,n, A1≤#B1 and A2≤B2, in which A=A1+A2 and B=B1+B2 are the C-N decompositions of A and B, respectivelyMatrix 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 A∈ℂm,n. Then, A can be written as(5)A=GAEA=EAHA,where EA∈ℂm,n is a partial isometry, i.e., EA∗=EA†, GA∈ℂm,m, and HA∈ℂn,n are Hermitian nonnegative definite matrices. The matrices EA, GA, and HA are uniquely determined by ℛEA=ℛGA and ℛEA∗=ℛHA, in which GA=A=AA∗1/2, HA=A∗=A∗A1/2, and EA=GA†A=AHA†.Based on the generalized polar decomposition, Hauke and Markiewicz [1] introduced the GL partial order: let A,B∈ℂm,n, and A=GAEA and B=GBEB be their polar decompositions, where ℛEA=ℛGA and ℛEB=ℛGB. Then,(6)A≤GLB⟺EA≤∗EBandGA≤LGB,⟺EA≤∗EBandHA≤LHB.After that, Wang and Liu [10] made the polar-like decomposition: let A∈ℂm,n. Then, A can be written as(7)A=GA12EAHA12,where EA, GA, and HA are given in ([3], Chapter 6, Theorem 7). On the basis of the polar-like decomposition, the WL partial order [10] is defined as(8)A≤WLB⟺GA12≤LGB12,EA≤∗EB,andHA12≤LHB12,in which A=GA1/2EAHA1/2 and B=GB1/2EBHB1/2 are the polar-like decompositions of A and B, respectively. In [11], Wang and Liu introduced the CL partial order:(9)A≤CLB⟺A○#A≤○#B○#BandA2A○#≤LB2B○#,in which A,B∈ℂnCM. 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 onℂnCM, 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.LetA∈ℂnCM, PA=AA†, and EA be as given in (5). Then,(10)A=FAQA,where(11)FA=UK000U∗,QA=UK−1ΣKK−1ΣL00U∗.We call (10) the P-2 expression of A. Furthermore, it is easy to check that(12)FA=EA○#A○#○#,QA=A○#EAA,(13)FA=AQA○#,A=QAQA○#A,FA∈ℂn,n is a EP-matrix, ρFA≤1, and QA∈ℂn,n.LetA,B∈ℂnCM. Consider the binary operation:(14)A≤LCB⇔FA≤LFBandQA≤○#QB,in which A=FAQA and B=FBQB are the P-2 expressions of A and B, 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 ℂnCM.Proof.
(1)
Reflexive: letA∈ℂnCM, and A=FAQA is the P-2 expression of A. We have(15)FA≤LFA,QA≤○#QA.So,A≤LCA.(2)
Antisymmetric: letA,B∈ℂnCM, A=FAQA and B=FBQB be their P-2 expressions. If A≤LCB and B≤LCA, that is, if(16)FA≤LFBandQA≤○#QB,FB≤LFAandQB≤○#QA.Then,A=B.(3)
Transitive: suppose thatA≤LCB and B≤LCC, that is, suppose(17)FA≤LFBandFB≤LFC,QA≤○#QBandQB≤○#QC.From the transitivity of the Löwner partial order and core partial order, we haveFA≤LFC and QA≤○#QC, that is, A≤LCC. By (1), (2), and (3), we know that the binary operation (14) is a partial order on ℂnCM.
Next, we give the characteristics of the LC partial order.Theorem 2.
LetA,B∈ℂnCM, rkB≥rkA≥1, A≤LCB, and A=FAQA, and B=FBQB be the P-2 expressions of A and B, respectively. Then, there exists a unitary matrix U such that(18)FA=UA1TA1−100000000U∗,QA=UTA1TA2TA3000000U∗,FB=UB1TA1−1B4TA1−1∗0B4TA1−1−B4TA1−1TA2TA4−1+B5TA4−10000U∗,QB=UTA1TA2TA30TA4TA5000U∗,B=UB1B2B3B4B5B6000U∗,where A1,B1∈ℂrkA,rkA, TA1 and TA4 are both nonsingular, TA2, TA3, TA4, and TA5 are arbitrary matrices with appropriate sizes, ρA1TA1−1≤1,(19)ρB1TA1−1,B4TA1−1∗,B4TA1−1,−B4TA1−1TA2TA4−1+B5TA4−1,≤1,B1−A1TA1−1,B4TA1−1∗,B4TA1−1,−B4TA1−1TA2TA4−1+B5TA4−1,≥0.Proof.
LetA∈ℂnCM. Applying (11), we get A=FAQA, where(20)FA=UKA000U∗,QA=UKA−1ΣAKAKA−1ΣALA00U∗.
Furthermore, we write(21)QA=UTA1TA2TA3000000U∗,FA=UA1TA1−100000000U∗,where TA1=KA−1ΣAKA,A1=KATA1 and TA2∈ℂrkA,rkB−rkA.
LetB∈ℂnCM,rkB≥rkA≥1, and B=FBQB be the P-2 expression of B. Since A≤LCB, by applying (14), we have QA≤○#QB. Then, by applying ([2], Lemma 3), we get(22)QB=UTA1TA2TA30TA4TA5000U∗,where TA4 is nonsingular. It is easy to check that(23)QB○#=UTA1−1−TA1−1TA2TA4−100TA4−10000U∗.
We write(24)B=UB1B2B3B4B5B6B7B8B9U∗.
Then, applying (13), we get B7=0,B8=0,B9=0, and(25)FB=BQB○#=UU∗BUTA1−1−TA1−1TA2TA4−100TA4−10000U∗=UB1TA1−1−B1TA1−1TA2TA4−1+B2TA4−10B4TA1−1−B4TA1−1TA2TA4−1+B5TA4−10000U∗.
SinceFA≤LFB and FB−FA≥0, then we get B4TA1−1=−B1TA1−1TA2TA4−1+B2TA4−1∗, and(26)FB−FA=UB1−A1TA1−1B4TA1−1∗0B4TA1−1−B4TA1−1TA2TA4−1+B5TA4−10000U∗≥0,that is,(27)B1−A1TA1−1B4TA1−1∗B4TA1−1−B4TA1−1TA2TA4−1+B5TA4−1≥0.
SinceρKA≤1, ρKB≤1, FA and FB are EP, and FA≤LFB, we obtain ρA1TA1−1≤1 and ρB1TA1−1B4TA1−1∗B4TA1−1−B4TA1−1TA2TA4−1+B5TA4−1≤1. Therefore, we have (18).
Next, we use Examples1 and 2 to explain the difference between LC partial order and minus (Löwner, GL, or CL) partial order.Example 1.
Let(28)A=U1200120130000000000U∗,B=U34003402300001200000U∗,U=1203200130223320−12002230−13.
Then,rkB=3, rkA=2, and(29)QA=U1001020000000000U∗,QB=U1001020000300000U∗,QA○#=U10000120000000000U∗,QB○#=U100001200001300000U∗,FA=U120000160000000000U∗,FB=U3400001300001600000U∗,B−A=U14001401300001200000U∗.
Since(30)FB−FA=U1400001600001600000U∗≥0⇒FA≤LFB,QAQA○#=U1000010000000000U∗=QBQA○#,QA○#QA=U1001010000000000U∗=QA○#QB,then QA≤○#QB. We have A≤LCB.(1)
SincerkB−A=3≠rkB−rkA=1, A is not below B under the minus partial order(2)
SinceB−A is not a positive semi-definite matrix, A is not below B under the Löwner partial orderExample 2.
Let(31)A=U15110405908−36611279115115−42519646676146400000000U∗,B=U1321517001223−14291656130112199444−198307700110011000000U∗,in which(32)U=1203200130223320−12002230−13.
Then,rkB=3, rkA=2, and(33)QA=U231632691398−977556−1316−1715701172351400000000U∗,QB=U231632691398−977556−1316−17157011723514001101100000U∗,QA○#=U1−100240000000000U∗,QB○#=U1−1157933024−131312570001000000U∗,FA=U25150015150000000000U∗,FB=U35150015310000011000000U∗.
Since(34)FB−FA=U150000110000011000000U∗≥0⟹FA≤LFB,QAQA○#=U1000010000000000U∗=QBQA○#,QA○#QA=U101579330−191637501−2243439187018900000000U∗=QA○#QB,so QA≤○#QB. We have A≤LCB.
(1) But(35)A○#=U10−1500−10300000000000U∗,B○#=U257−407−259551080−1071007−106012400010000000U∗,A2A○#=U15110001151150000000000U∗,B2B○#=U1321561344101301126101361000110000000U∗,B2B○#−A2A○#=U2151306134410−1301606101361000110000000U∗.
B2B○#−A2A○# is not a positive semidefinite matrix, so A is not below B under the CL partial order.
(2) Since(36)AA∗=U1120622413−1152412200−115241223597610000000000U∗,BB∗=U2022123736198713403540361987474851612495630134035412495631352395400000U∗,AB∗=U3905536474834473379790−215541−12238576113860000000000U∗,AA∗1/2BB∗1/2=U557683159487815503780174910616335255817328289000000000U∗,AB∗≠AA∗1/2BB∗1/2,A is not below B under the GL partial order.Theorem 3.
LetA,B∈ℍ≥n, rkA=rkB≥1. Then,(37)A≤LCB⟺A≤LB.Proof.
WhenrkA=rkB, we have(38)QA=QB=UTA1TA2TA3000000U∗.
Then,(39)FA=AQA○#=UA1TA1−100000000U∗,FB=BQB○#=UB1TA1−100000000U∗,FB−FA=UB1−A1TA1−100000000U∗.
Therefore,A≤LCB⟺FA≤LFB⟺A≤LB.Corollary 1 (see [7]).
LetA,B∈ℍ≥n. Then, A≤CLB if and only if A≤LB.In ([8], Theorem 8.5.15), Mitra has verified that when A,B∈ℍ≥n, there has A≤GLB if and only if A≤LB. By applying Theorem 3, Corollary 1, and ([8], Theorem 8.5.15), we get Corollary 2.Corollary 2.
LetA,B∈ℍ≥n and rkA=rkB≥1. Then,(40)A≤LCB⇔A≤CLB⇔A≤GLB.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 bothA and B are EP.Theorem 4.
LetA,B be EP, rkB>rkA≥1, A≤LCB, A=FAQA, and B=FBQB be the P-2 expression of A and B. Then, there exists a unitary matrix U such that(41)FA=UIrkA00000000U∗,QA=UTA100000000U∗,FB=UB1TA1−1B2TA4−10B2TA4−1∗B5TA4−10000U∗,QB=UTA1000TA40000U∗,where B1∈ℂrkA,rkA, ρB1TA1−1B2TA4−1B2TA4−1∗B5TA4−1≤1, TA1 and TA4 are both nonsingular, and B1−TA1TA1−1B2TA4−1B2TA4−1∗B5TA4−1≥0.Proof.
LetA,B be EP, rkB≥rkA≥1, A≤LCB. There exists a unitary matrix U such that(42)A=UTA1000U∗=FAQA=UΣKAΣLA00U∗=UKA000U∗UKA−1ΣKAKA−1ΣLA00U∗.
Then,LA=0, KA=IrkA. Furthermore,(43)FA=UIrkA00000000U∗,QA=UΣA000U∗=UTA100000000U∗.
SinceB is EP,(44)B=UB1B20B2∗B50000U∗and rkB≥rkA≥1, we get(45)QB=UTA1000TA40000U∗,QB○#=UTA1−1000TA4−10000U∗.
Then,(46)FB=BQB○#=UU∗BQB○#=UU∗BUTA1−1000TA4−10000U∗=UB1B20B2∗B50000TA1−1000TA4−10000U∗=UB1TA1−1B2TA4−10B2∗TA1−1B5TA4−10000U∗.
SinceρKA≤1, ρKB≤1, and FA and FB are both EP, we have B2∗TA1−1=B2TA4−1∗, and ρB1TA1−1B2TA4−1B2TA4−1∗B5TA4−1≤1. And because FA≤LFB and FB−FA≥0, then(47)FB−FA=UB1−TA1TA1−1B2TA4−10B2TA4−1∗B5TA4−10000U∗≥0.
Therefore,(48)B1−TA1TA1−1B2TA4−1B2TA4−1∗B5TA4−1≥0.
So, we have (41).
It is noteworthy thatA≤LCB⇏A≤LB, when A and B both are EP.Example 3.
Let(49)A=13000−120000,B=14000−140000,rkB=rkA=2, and(50)QA=13000−120000=QB,QA○#=3000−20000=QB○#.
We obtainQB≤○#QA. Since(51)FA=100010000,FB=34000120000,FA−FB=14000120000≥0⟹FBL≤FA.
We haveB≤LCA. But(52)A−B=112000−140000≱0.
B is not below A under the Löwner partial order. So, B≤LCA⇏B≤LA.



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*Source: 3215038-2020-09-09.xml*