An eigenvalue inequality of a class of matrices and its applications in proving the Fischer inequality

ZHANG Huamin, YIN Hongcai

(1.Department of Mathematics & Physics, Bengbu University, Bengbu 233030,Anhui Province, China;2.School of Management Science and Engineering, Anhui University of Finance & Economics,Bengbu 233000, Anhui Province, China)

An eigenvalue inequality of a class of matrices and its applications in proving the Fischer inequality

ZHANG Huamin1, YIN Hongcai2

(1.Department of Mathematics & Physics, Bengbu University, Bengbu 233030,Anhui Province, China;2.School of Management Science and Engineering, Anhui University of Finance & Economics,Bengbu 233000, Anhui Province, China)

The Hadamard inequality and Fischer inequality play an important role in the matrix study. Many articles have addressed these inequalities providing new proofs, noteworthy extensions, generalizations, refinements, counterparts and applications. This paper discusses the eigenvalues of a class of matrices related to the real symmetric positive definite matrix and establishes an inequality of the eigenvalues. By using this inequality, the Fischer determinant inequality and Hadamard determinant inequality are proved.

positive definite matrix; eigenvalue; eigenvector; determinant inequality

1 Introduction and preliminaries

Inequality is an active research topic in recent years,the classical convexity has been generalized and extended in a diverse manner.One of them is the pre-invexity,introduced by WEIR et al[1]as a significant generalization of convex functions.Many researchers have studied the basic properties of the pre-invex functions and their role in optimization,variational inequalities and equilibrium problems[2-4].

Hadamard and Fischer inequalities are prima-ry inequalities for the real symmetric positive def-inite matrix, and there are many inequalities can be proved by using these two inequalities.There are many methods to prove these inequalities[5-7].Some results have been established inspired by the Hadamard inequality[8-9].

The real symmetric positive definite matrix has many properties and has been used in many ar-eas[10-12].Some properties can be used to prove the Hadamard inequality.In this note,inspired by the results established in[13-14],a new eigenvalue inequality related to the real symmetric positive definite matrix is proposed, and the Hadamard and Fischer inequality are proved by using this new inequality.

Firstly, let us introduce some notations and lemmas.Inis the identity matrix with ordern×n.For a square matrixA,we use λ[A],det(A) andATrepresent the set of the eigenvalues,the deter-minant and the transpose ofA,respectively.

Next,we introduce two lemmas.The following result about the block matrix determinant is well known[10].

Lemma1If matrixAis invertible,then for any block matrix,we have

(1)

or if marixDis invertible,then we have

(2)

Lemma2IfA∈Rm×nis a full column-rank matrix,theA(ATA)-1ATis idempotent and the eigenvalues ofA(ATA)1ATare 1 or 0, there exists an orthogonal matrixQsuch that

Q[A(ATA)-1AT]Q=diag[1,…,1,0,…,0]=∶Λ.

Furthermore,we have rank [A]=n.

This lemma was suggested in[14],for convenience,we give the proof here.

ProofIf σ∈λ[A(ATA)1AT],then there exists a nonzero vectorx∈Rm, satisfying

A(ATA)1ATx=σx.

Thus,we have

[A(ATA)-1ATx]T[A(ATA)-1ATx]=(σx)T(σx)，
xT[A(ATA)-1AT][A(ATA)1AT]x=σ2‖x‖2，
xT[A(ATA)-1ATA(ATA)1AT]x=σ2‖x‖2，
xT[A(ATA)1AT]x=σ-2‖x‖2，
xTσx=2‖x‖2，
σ‖x‖2=σ2‖x‖2.

Since‖x‖2≠0,A(ATA)-1AThas eigenvaluesσ=0 orσ=1.Because of the symmetry ofA(ATA)-1AT,there exists a real orthogonal matrixQ:=[q1，q2,…,qm]∈Rm×msuch that

QT[A(ATA)-1AT]Q= diag[1,…,1,0,…,0]=Λ.

On the other hand,since (ATA)-1ATis the left pseudo-inverse ofA,we have

rank[A]=rank[QT[A(ATA)-1AT]Q]=
rank[A(ATA)-1AT]=rank[A]=n.

This proves lemma 2.

2 An inequality of a class of matrices

In this section,we will establish a new property about the eigenvalues related to the symmetric positive definite matrix.IfA∈Rn×nis a symmetric positive definite matrix,then there exists an invertible matrixBsuch thatA=BBT.Suppose thatBcan be expressed as a block matrix

Set

With these symbols,the following result holds.

Theorem1If the eigenvalues of the matrixN-1Aareδ1,δ2,…,δn,then 0<δ1δ2…δn≤1.

ProofLetf(λ)∶=det(λIn-N-1A)be the characteristic polynomial of matrixN-1A,we have

f(λ)=

(3)

We verify that 2 not belongs to the eigenvalues of the matrixN-1A.If 2 is the eigenvalue of the matrixN-1A,then

On the other hand,

f(2)=

This is a contradiction,so 2 is not a eigenvalue of the matrixN-1A.

According to lemma 1,suppose thatm≥p,equation (3) can be manipulated as

det((λ-1)Im)det((λ-1)Ip-(λ-1)-1Ip×

(4)

[q1,q2,…,qn]diag[1,…,1,0,…,0]=

[q1,q2,…,qm,0,0,…,0],

(5)

(6)

(7)

(k1q2+k2q2+…+knqn)=

(8)

(k1q1+k2q2+…+knqn)=

k1q1+k2q2+…+kmqm.

(9)

The both side of equation (6) multiply byTgives

=.

(10)

According to equations(7) and (9),the left-side of equation (10) can be rewritten as

(k1q1+k2q2+…+knqn)T×

(k1q1+k2q2+…+kmqm)=

(11)

Combining equations(10)(8)and(11)gives

Hence,we have

(12)

(13)

whereRis strictly upper triangular.Substituting equation(13)into equation(4)and simplifying it, give

From this equation,we can see that the eigenvalues ofN-1Aare

1+ρ1,1-ρ1,…,1+ρp,1-ρp,1,…,1.

(14)

From this equation,we have

0≤δ1δ2…δn=

(1+ρ1)(1-ρ1)…(1+ρp)(1-ρp)=

(15)

Since 2 is not the eigenvalue of the matrixN-1A,an improvement of inequality (12) is 0≤ρ<1.

Correspondingly,inequality (15) can be im-proved as

0≤δ1δ2…δn=

(1+ρ1)(1-ρ1)…(1+ρp)(1-ρp)=

The proof is completed.

Remark1The above proof shows that the sup-positionm≥pis not essential.In fact,ifm

f(λ)=

det((λ-1)Ip)det((λ-1)Im-(λ-1)-1×

This manipulation does not change the subsequent proof.

3 New proof of the Fischer inequality

In this section,we will use the results in theo-rem 1 to prove two determinant inequalities related to the symmetric positive definite matrix,that is,the Fischer inequality and the Hadamard inequali-ty.

Theorem2Considering the following symmetric positive definite block matrix

here Mii,i=1,2,…,k,are the definite submatri-ces,then

det(M)≤det(M11)det(M22)…det(Mkk).

det(N-1M)≤det(N-1)det(M)=δ1δ2…δn≤1.

That is,

det(M)≤det(N)=det(M11)det(M22).

Fork>2,using this manipulation successively gives

det(M)≤det(M11)det(M22)…det(Mkk).

The proof is completed.

It is clear that Hadamard inequality is the spe-cial case of Fischer inequality whenk=n,so the following inequality holds.

Theorem3IfM=(mij) ∈Rn×nis a symmetric positive definite matrix,then

det(M)≤m11m22…mnn.

4 Conclusions and future work

The eigenvalues of a class of matrices related to the real symmetric positive definite matrix are discussed in this paper, and an inequality about the eigenvalues is established.Using this result,the Fischer inequality and the Hadamard inequality of the positive definite matrix are proved.

SandTdenote the subsets of set W:= {1,2,…,n}and S and T satisfy S ∪T=W.cd(S) denotes the cardinality of set S.Screpresents the complementary set of S.MSdenotes the principal submatrix determined by set S.

Consider the following of the Koteljanskii,Fan and Szasz inequalities[15],

j=1,2,…,n-1,

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張華民1,殷紅彩2

( 1.蚌埠學院 數理系,安徽 蚌埠 233030; 2.安徽財經大學 管理科學和工程學院,安徽 蚌埠 233000)

Hadamard和Fischer不等式在矩陣研究中起重要作用.已有大量文獻研究此兩不等式的新證明、 推廣、 細化及應用.本文研究了和實對稱正定矩陣相關的一類矩陣的特征值，并建立了關于這類矩陣特征值乘積范圍的一個不等式,利用此不等式證明了行列式的Fischer和Hadamard不等式.

正定矩陣;特征值;特征向量;行列式不等式

O 151.2

：A

：1008-9497(2017)05-511-05

date：Feb.4,2016.

Supported by Natural Science Foundation of Anhui Provincial Education Department (KJ2016A458) and Excellent Personnel Domestic Visiting Project (gxfxZD2016274).

Abouttheauthor:ZHANG Huamin (1972-),ORCID:http://orcid.org/0000-0002-7416-7415,male,doctor,associate professor,the field of interest are matrix theory and its applications,E-mail:zhangeasymail@126.com.

10.3785/j.issn.1008-9497.2017.05.002

一類矩陣特征值的不等式及其在Fischer不等式證明中的應用.浙江大學學報(理學版),2017,44(5):511-515