# Fischer Type Log-Majorization of Singular Values on Partitioned Positive Semidefinite Matrices

Authors: Benju Wang; Yun Zhang
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we establish a Fischer type log-majorization of singular values on partitioned positive semidefinite matrices, which generalizes the classical Fischer's inequality. Meanwhile, some related and new inequalities are also obtained.


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## Body

## 1. Introduction

We denote byMn,m the vector space of all complex n×m matrices. In particular, when n=m, Mn represents the set of square matrices of order n. Let A and B be Hermitian matrices. We use the notation A≤B or B≥A to mean that B−A is positive semidefinite. Particularly, B≥0 (B>0) means that B is positive semidefinite (B is positive definite). The singular values of A∈Mn are the nonnegetive square roots of the eigenvalues of A∗A. We denote that sjA is the jth largest singular value of A∈Mn and we denote sA=s1A,s2A,⋯,snA. Denote by ∥·∥∞ the spectral norm. For A∈Mn, it is evident that ∥A∥∞=s1A.We rearrange the components ofx=x1,x2,⋯,xn∈ℝn in decreasing order as x1≥x2≥⋯≥xn. Let x=x1,⋯,xn,y=y1,⋯,yn∈ℝn. If
(1)∑i=1kxi≤∑i=1kyi,k=1,2,⋯,n,then we say that x is weakly majorized by y and denotes x≺ωy. If x≺ωy and ∑i=1nxi=∑i=1nyi, then we say that x is majorized by y and denotes x≺y. Let nonnegative vectors x,y∈ℝ+n. If
(2)∏i=1kxi≤∏i=1kyi,k=1,2,⋯,n,then we say that x is weakly log-majorized by y and denotes x≺ωlogy. If x≺ωlogy and ∏i=1nxi=∏i=1nyi, then we say that x is log-majorized by y and denotes x≺logy.A norm onMn is called unitarily invariant if ∥UAV∥=∥A∥ for any A∈Mn and any unitary U,V∈Mn. For A∈Mn and 1≤k≤n, the kth compound matrix of A is denoted by CkA. We list one of the useful properties on compound matrices: for A,B∈Mn, CkAB=CkACkB. A series of properties of compound matrices can be seen in [1–3]. A complex matrix C is called contraction if C∗C≤I or equivalently s1C≤1. We denote the block matrix
(3)A00C,by A⊕C.The well-known Fischer’s inequality for determinant on the partitioned positive semidefinite matrix is the following.Let(4)H=ABB∗C≥0,withA∈MpandC∈Mq.Then(5)detH≤detAdetC=detA⊕C.Let the eigenvalues ofA∈Mn be λ1,λ2,⋯,λn with ∣λ1∣≥∣λ2∣≥⋯≥∣λn∣. Weyl [4] proved that ∣λi∣i=1n≺logsA. Since for positive semidefinite matrices, singular values and eigenvalues are the same. Fischer’s inequality can be rewritten as the following.
(6)detH=∏i=1p+qsiH≤∏i=1psiA∏i=1qsiC=∏i=1p+qsiA⊕C=detA⊕C.The motivation of this paper is to give the log-majorization relationship between the singular values of partitioned positive semidefinite matrix and its main diagonal matrix, which generalizes classical Fischer’s inequality. In addition, we will also establish some singular value inequalities between partitioned positive semidefinite matrix and its main diagonal matrix.


## 2. Main Results

First, we list some lemmas that are used in our proofs.Lemma 1 (Zhan [1], p. 71).
LetC be a complex matrix (not necessary square). Then, C is a contraction if and only if
(7)ICC∗I≥0.Lemma 2 (Zhan ([1], Theorem 3.34).
(8)ABB∗C≥0,if and only if A≥0,C≥0, and there exists a contraction W such that B=A1/2WC1/2.Lemma 3 (Zhan [1], p. 80).
If the singular values ofA∈Mn are s1,⋯,sn, then the eigenvalues of
(9)φA≔0AA∗0,are s1,⋯,sn,−sn,⋯,−s1.More general, by Jordan-Wielandt theorem ([5], Theorem 4.2), for A∈Mp,qp≥q, the matrix
(10)0AA∗0,has eigenvalues ±si,0,⋯,0⏞p−qtimes,i=1,⋯,q.Applying Lemma3, we obtain the following.Lemma 4.
LetW∈Mp,qp≥q be the contractive matrix. Then,
(11)sIWW∗I=sI+φW=1+s1,⋯,1+sq,1,⋯,1⏞p−qtimes,1−sq,⋯,1−s1,where si is the ith largest singular value of W.Proof.
W is a contractive matrix if and only if
(12)IWW∗I≥0.
For positive semidefinite matrices, singular values and eigenvalues are the same. Note that(13)IWW∗I=I00I+0WW∗0,by Lemma 3 and spectral mapping theorem, this completes the proof.Next, we will establish the log-majorization relationship between the singular values of partitioned positive semidefinite matrix and its main diagonal matrix.Theorem 5.
Let(14)ABB∗C∈Mp+q,be positive semidefinite matrix with A∈Mp,C∈Mqp≥q. Then, there exists a contractive matrix W∈Mp,q such that
(15)sABB∗C≺logsiI+φWsiA⊕Ci=1p+q.Proof.
Applying Lemma2, there exists a contractive matrix W such that B=A1/2WC1/2. Then,
(16)ABB∗C=A1/200C1/2∗IpWW∗IqA1/200C1/2.
Denote(17)H=ABB∗C,Q=IpWW∗Iq,P=A1/200C1/2,T=A⊕C.
The equality (16) can be rewritten as H=P∗QP. Using compound matrices and the fact that for any square matrix A, ∥A∥∞2=∥A∗A∥∞, for 1≤k≤p+q, we have
(18)∏j=1ksjH=∏j=1ksjP∗QP=s1CkP∗QP=s1CkP∗CkQCkP=CkP∗CkQCkP∞≤CkP∗∞CkQ∞CkP∞=CkP∗∞2CkQ∞=CkP∗CkP∞CkQ∞=CkP∗P∞CkQ∞=CkT∞CkQ∞=∏j=1ksjTsjQ.
In addition, fork=p+q, we have
(19)detH=∏j=1p+qsjH=detP∗QP=detQT=∏j=1p+qsjI+φW∏i=1p+qsiA⊕C=∏j=1q1−sjW2∏i=1p+qsiA⊕C,where the last equality is due to Lemma 4, this completes the proof.Corollary 6.
Let(20)H=ABB∗C∈Mp+q,be positive semidefinite matrix with A∈Mp,C∈Mqp≤q. Then, there exists a contractive matrix W∈Mp,q such that
(21)detH=∏j=1q1−sjW2detA⊕C.Remark 7.
From the equality (21), we can see that detH≤detA⊕C. The equality holds if and only if sW=0, i.e., W=0; it also implies that B=0.Corollary 8.
Let(22)H=ABB∗C∈Mp+q,be positive semidefinite matrix with A∈Mp,C∈Mq. Then, there exists a contractive matrix W∈Mp,q such that
(23)ABB∗C≤1+s1W·A⊕C.for all unitarily invariant norms ·.Proof.
Recall that weak log-majorization is stronger than weak majorization, i.e.,x≺ωlogy implies x≺ωy. By Theorem 5, we have
(24)s1H,⋯,sp+qH≺ω1+s1Ws1A⊕C,⋯,1−s1Wsp+qA⊕C.
It is clear that(25)1+s1Ws1A⊕C,⋯,1−s1Wsp+qA⊕C≤1+s1Ws1A⊕C,⋯,1+s1Wsp+qA⊕C.
Therefore, we have(26)s1H,⋯,sp+qH≺ω1+s1Ws1A⊕C,⋯,1+s1Wsp+qA⊕C.
Using the Fan Dominance Principle (see [6]), this completes the proof.In the following section, we will establish some singular value inequalities between partitioned positive semidefinite matrix and its main diagonal matrix. The following well-known result is due to Ky Fan [1].Lemma 9.
LetA,B∈Mn,1<i,j≤n,i+j−1≤n. Then,
(27)si+j−1AB≤siAsjB.
In particular,(28)sjAB≤s1AsjB,sjAB≤s1BsjA,j=1,⋯,n.Theorem 10.
Let(29)ABB∗C∈Mp+q,be positive semidefinite matrix with A∈Mp,C∈Mqp≥q. Then, there exists a contractive matrix W∈Mp,q such that
(30)siABB∗C≤1+s1WsiA⊕C,i=1,⋯,p+q.
In addition, for1≤i≤q,(31)siABB∗C≤1+siWs1A⊕C,for q+1≤i≤q+p,(32)siABB∗C≤s1A⊕C.Proof.
(33)siABB∗C=siA1/200C1/2∗IpWW∗IqA1/200C1/2≤siA1/200C1/2s1IpWW∗IqA1/200C1/2≤siA1/200C1/2s1IpWW∗IqsiA1/200C1/2=s1IpWW∗IqsiA00C=1+s1WsiA00C.
In (33), change the position of
(34)A1/200C1/2,and
(35)IpWW∗IqA1/200C1/2,
By similar proof method, we can get the following inequality.(36)siABB∗C≤siIWW∗Is1A⊕C,i=1,⋯,p+q.
Using Lemma4, we can divide into the following cases:
Case 1.1≤i≤q. Then,
(37)siABB∗C≤1+siWs1A⊕C.
Case 2.q+1≤i≤p+q. Then,
(38)siABB∗C≤s1A⊕C.
This completes the proof.The following theorem is proved by using the result given by Hirzallah and Kittaneh [7].Lemma 11.
LetA,B∈Mn and 0<α<1. Then,
(39)sjαA+1−αB≤sjA⊕B,j=1,⋯,2n.
In particular, lettingα=1/2, we have
(40)sjA+B2≤sjA⊕B,j=1,⋯,2n.Theorem 12.
Let(41)ABB∗C∈Mp+q,be positive semidefinite matrix with A∈Mp,C∈Mqp≥q. Then, for j=1,⋯,p+q,
(42)sjA00C≤sjABB∗C⊕ABB∗C.Proof.
Since(43)Ip00−IqABB∗CIp00−Iq=A−B−B∗C,we have
(44)sjABB∗C=sjA−B−B∗C.
Meanwhile(45)A00C=12ABB∗C+12A−B−B∗C.
Applying Lemma11 to (45), we have
(46)sjA00C≤sjABB∗C⊕A−B−B∗C=sjABB∗C⊕ABB∗C.
This complete the proof.Remark 13.
One may ask whether the inequality holds(47)sjA00C≤sjABB∗C,j=1,⋯,p+q.
But it needs not be true. For example, setting(48)ABB∗C=1111≥0,A00C=1001,
Then,(49)s1111=2,0,s1001=1,1.
Obviously,(50)s21111=0<1=s21001.



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