# A Characterization on Singular Value Inequalities of Matrices

Authors: Wei Dai; Yongsheng Ye
Journal: Journal of Function Spaces
 (2020)

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## Abstract

We obtain a characterization of pair matricesA and B of order n such that sjA≤sjB,j=1,…,n, where sjX denotes the j-th largest singular values of X. It can imply not only some well-known singular value inequalities for sums and direct sums of matrices but also Zhan’s result related to singular values of differences of positive semidefinite matrices. In addition, some related and new inequalities are also obtained.


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## Body

## 1. Introduction

LetMn denote the vector space of all complex n×n matrices, and let Hn be the set of all Hermitian matrices of order n. We always denote the eigenvalues of A∈Hn in decreasing order by λ1A≥λ2A≥⋯≥λnA. For A,B∈Hn, we use the notation A≤B or B≥A to mean that B−A is positive semidefinite. Clearly, “≤” and “≥” define two partial orders on Hn, each of which is called Löwner partial order. In particular, B≥O (res., B>O) means that B is positive semidefinite (res., B is positive definite). For T∈Mn, the singular values of T, denoted by s1T,s2T,…,snT, are the eigenvalues of the positive semidefinite matrix T=T∗T1/2, enumerated as s1T≥s2T≥⋯≥snT and repeated according to multiplicity. It follows that the singular values of a normal matrix are just the moduli of its eigenvalues. In particular, if T∈Mn is positive semidefinite, then singular values and eigenvalues of T are the same. For more information on this related topic, see [1–3]. Recall that a complex matrix C∈Mn is called contraction if C∗C≤I, or equivalently C≤1, where ⋅I denotes the spectral norm, the identity matrix of order n, respectively.Here, we denote the block matrixA00B by A⊕B.This paper intends to give a characterization of pair matricesA,B∈Mn such that sjA≤sjB,j=1,…,n. It can generalize some singular value inequalities for sums and direct sums of matrices due to Hirzallah and Kittaneh [4]. Using this characterization, we give a new proof of Zhan’s result related to singular values of differences of positive semidefinite matrices [5]. Several applications of this characterization are presented, and some related and new inequalities are also obtained.


## 2. Main Results

The following well-known results are due to Ky Fan.Lemma 1.
LetA,B∈Mn,1<i,j≤n,andi+j−1≤n. Then(1)si+j−1AB≤siAsjB.
In particular,(2)sjAB≤AsjB,sjAB≤BsjA,j=1,…,n,where ⋅ denotes the spectral norm.Next, we will prove the following useful result.Theorem 1.
LetA,B∈Mn. Then,(3)sjA≤sjB,j=1,2,…,n,if and only if contractive matrices W and V exist such that A=WBV. In particular, suppose A and B are positive semidefinite. Then,(4)sjA≤sjB,j=1,2,…,n,if and only if there exists a contractive matrix W such that A=WBW∗.Proof.
The sufficiency follows from Lemma1. We prove the necessity.
Consider the singular value decompositionsA and B. Then, unitary matrices U1,U2,V1, and V2 exist such that(5)U1AV1=diags1A,s2A,…,snA,U2BV2=diags1B,s2B,…,snB.
Setting(6)βi=1,siB=0,siAsiB,siB≠0.
DenoteW0=diagβ1,β2,…,βn. It is clear that O≤W0≤I. Then,(7)A=U1∗W0U2BV2W0V1∗.
LetW=U1∗W0U2 and V=V2W0V1∗. It is trivial that both W and V are contractive. Then, A=WBV.
LetA and B be positive semidefinite. Note that for positive semidefinite matrices, singular values and eigenvalues are the same. Using the spectral decompositions of A and B, the unitary matrices U1 and U2 exist such that(8)U1∗AU1=diags1A,s2A,…,snA,U2∗BU2=diags1B,s2B,…,snB.
The left proof is similar to the above. This completes the proof.
It is clear that forA∈Mn,(9)sjA=sjA⊕O,j=1,2,…,n,where O is a zero square matrix of any order. An interesting consequence of Theorem 1 is the following.Corollary 1 (see [4], Theorem 2.1).
LetAi,Bi,Xi∈Mn,i=1,…,m. Then,(10)sj∑i=1mAiXiBi≤∑i=1mAi∗21/2∑i=1mBi21/2sj⊕i=1mXi,for j=1,…,n.Proof.
Denote(11)A=A1A2⋯Am00⋯0⋮⋮⋱⋮00⋯0,B=B10⋯0B20⋯0⋮⋮⋱⋮Bm0⋯0,X=⊕i=1mXi.
Then,A=∑i=1mAi∗21/2andB=∑i=1mBi21/2. Then,(12)AAXBB=1AB∑i=1mAiXiBi⊕O,where O is the zero matrix of order nm−1. It is clear that both 1/AA and 1/BB are contractive matrices. By Theorem 1, this completes the proof.
The following result generalizes the result due to Hirzallah and Kittaneh ([4], Corollary 2.1).Theorem 2.
LetA1,A2,…,Am∈Mn, and let λ1,λ2,…,λm be complex numbers such that λ1+λ2+⋯+λm≤1. Then, for j=1,…,n,(13)sjλ1A1+λ2A2+⋯+λmAm≤sjA1⊕A2⊕⋯⊕Am.Proof.
Note that eachλk=λkeiθk,k=1,…,m, for some θk∈0,2π. Denote(14)A=λ1Inλ2In⋯λmIn00⋯0⋮⋮⋱⋮00⋯0,B=λ1eiθ1In0⋯0λ2eiθ2In0⋯0⋮⋮⋱⋮λmeiθmIn0⋯0,X=⊕i=1mAi.
Then,(15)AXB=λ1A1+λ2A2+⋯+λmAm⊕Onm−n,with contractive matrices AandB. By Theorem 1, this completes the proof.Remark.
This result admits the following important special cases:(1)
LetA1,A2,…,Am∈Mn, and let λ1,λ2,…,λm be nonnegative real numbers with λ1+λ2+⋯+λm=1. Then,(16)sjλ1A1+λ2A2+⋯+λmAm≤sjA1⊕A2⊕⋯⊕Am,j=1,…,n.(2)
LetA,B∈Mn. Then,(17)sjA+iB2≤sjA⊕B,j=1,…,n.
In particular, lettingB=iA∗, we have(18)sjImA≤sjA⊕A,j=1,…,n,which can be regarded as a complement to Hirzallah and Kittaneh’s inequality (see ([4], Corollary 2.3)), which says(19)sjReA≤sjA⊕A,j=1,…,n,where ReA and ImA denote the real part and the imaginary part of A, respectively.
Applying Theorem2, we have the following result.Corollary 2.
LetA∈Mn. Then,(20)sjA≤sjA+A∗⊕A−A∗,j=1,…,n.Proof.
Considering the Cartesian decomposition ofA,A=ReA+iImA. By Theorem 2, we have for 1≤j≤n,(21)sjA=2sjReA+iImA2≤2sjReA⊕ImA=sjA+A∗⊕A−A∗.
The last equality on the above is due to the fact that singular values are unitarily invariant:sjUAV=sjA,j=1,…,n, for every A and all unitary UandV.Remark.
Next, we show that equality is possible in inequality (20) for some nonzero square matrix T. Consider the matrix T=0100. Then,(22)T+T∗=0110,T−T∗=01−10.
It is clear thats1T=s1T+T∗⊕T−T∗=1.
The well-known arithmetic-geometric mean inequality for singular values due to Bhatia and Kittaneh [6] says that if A,B∈Mn, then(23)2sjAB∗≤sjAA∗+BB∗,j=1,…,n.
Zhan [5] asserts that for A,B∈Mn, if A≥O and B≥O, then(24)sjA−B≤sjA⊕B,j=1,…,n.
In [7], Tao has proved that if A,B,C∈Mn such that ABB∗C≥O, then(25)sjB≤12sjABB∗C,j=1,…,n.
It has been pointed out [7] that the three inequalities (23)–(25) are equivalent.
Recently, Audeh and Kittaneh [8] proved that let A,B,C∈Mn. If ABB∗C≥O, then(26)sjB≤sjA⊕C,j=1,…,n.
Next, we will interpolate Audeh and Kittaneh’s inequality (26) by proving.Theorem 3.
LetA,B,C∈Mn such that ABB∗C≥O. Then,(27)sjB≤12sjABB∗C≤sjA⊕C,j=1,…,n.Proof.
SinceABB∗C≥O, we have(28)ABB∗C=MN∗MN,for some M,N∈M2n,n. Then,(29)A=M∗M,C=N∗N.
Note that singular values and eigenvalues of positive semidefinite matrices are the same. By Theorem 1.27 in [3], for 1≤j≤n, then(30)sjABB∗C=sjMN∗MN=sjMNMN∗.
Appling Theorem2 and Theorem 1.27 in [3], we obtain that for 1≤j≤n,(31)sjMM∗+NN∗≤2sjMM∗⊕NN∗=2sjM∗M⊕N∗N,i.e.,(32)12sjABB∗C≤sjA⊕C.
Combining inequality (25) by Tao and the above inequality gives (27).
LetA=Udiagλ1,λ2,…,λnU∗ be the spectral decomposition of A∈Hn with U unitary and λ1≥⋯≥λp≥0>λp+1≥⋯≥λn. Denote(33)A+=Udiagλ1,…,λp,0,…,0U∗,A−=Udiag0,…,0,−λp+1,…,−λnU∗.
ThenA+≥O, A−≥O, and A=A+−A−. This is called the Jordan decomposition of A. As is known, there are at least four different proofs of Zhan’s result [5, 9]. Applying Theorem 1 and the Jordan decomposition, we give another direct proof of Zhan’s inequality (8).Proof.
DenoteT=A−B∈Hn. Considering the Jordan decomposition of T, T=T+−T−. Recall from ([3], Theorem 3.6) that if T=A−B with A≥O,B≥O, then(34)sjT+≤sjA,sjT−≤sjB,j=1,…,n.
By Theorem 1, there exist two contractive matricesW1 and W2 such that(35)T+=W1∗AW1,T−=W2∗AW2.
DenoteW=W1⊕W2. Then, W is a contractive matrix and so T+⊕T−=W∗A⊕BW. For T∈Hn,(36)sjT=sjT⊕O=sjT+⊕T−,j=1,…,n,where O∈Mn. By Theorem 1, this completes the proof. [10].



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*Source: 1657381-2020-02-18.xml*