Some new inequalities for the minimum eigenvalue of M matrices

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R E S E A R C H

Open Access

Some new inequalities for the minimum

eigenvalue of

M

-matrices

Feng Wang

*

_{and De-shu Sun}

*_{Correspondence:}

[email protected]

College of Science, Guizhou Minzu University, Guiyang, Guizhou 550025, P.R. China

Abstract

Some new inequalities for the minimum eigenvalue ofM-matrices are obtained. These inequalities improve existing results, and the estimating formulas are easier to calculate since they only depend on the entries of matrices. Finally, some examples are also given to show that the bounds are better than some previous results.

MSC: 15A18; 15A42

Keywords: M-matrix; nonnegative matrix; Hadamard product; spectral radius; minimum eigenvalue

1 Introduction

LetCn×n₍_Rn×n_{) denote the set of all}_n_×_n_{complex (real) matrices,}_A_{= (}_a_ij)_∈_Cn×n_,_N₌ {, , . . . ,n}. We writeA≥ if allaij≥ (i,j∈N).Ais called nonnegative ifA≥. LetZn denote the class of alln×nreal matrices all of whose oﬀ-diagonal entries are nonpositive. A matrixAis called anM-matrix [] ifA∈Znand the inverse ofA, denoted byA–, is nonnegative.Mnwill be used to denote the set of alln×n M-matrices.

LetAbe anM-matrix. Then there exists a positive eigenvalue ofA,τ(A) =ρ(A–₎–_,

whereρ(A–) is the spectral radius of the nonnegative matrixA–,τ(A) =min{|λ|:λ∈ σ(A)},σ(A) denotes the spectrum ofA.τ(A) is called the minimum eigenvalue ofA[, ]. The Hadamard product of two matricesA= (aij)∈Rn×n_and_B_{= (}_b_ij)_∈_Rn×n_{is the} ma-trixA◦B= (aijbij)∈Rn×n.

Ann×nmatrixAis said to be reducible if there exists a permutation matrixPsuch that

PTAP=

A 

A A

,

whereA,Aare square matrices of order at least one. We callAirreducible if it is not

reducible. Note that any nonzero × matrix is irreducible.

Estimating the bounds for the minimum eigenvalueτ(A) of anM-matrixAis an interest-ing subject in matrix theory, and it has important applications in many practical problems [–]. Hence, it is necessary to estimate the bounds forτ(A).

In [], Shivakumaret al.obtained the following bound forτ(A): LetA= (aij)∈Rn×nbe a weakly chained diagonally dominantM-matrix. Then

r(A)≤τ(A)≤R(A), τ(A)≤min i∈N aii,



M≤τ(A)≤ 

m. ()

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Subsequently, Tian and Huang [] provided a lower bound forτ(A) by using the spectral radius of the Jacobi iterative matrixJAofA: LetA= (aij)∈Rn×nbe anM-matrix andA–= (αij). Then

τ(A)≥   + (n– )ρ(JA)

 maxi∈N{αii}

. ()

Recently, Liet al.[] improved () and gave the following result: LetB= (bij)∈Rn×n_be anM-matrix andB–_{= (}_β_{ij). Then}

τ(B)≥ 

maxi=j{βii+βjj+ [(βii–βjj)+ (n– )βiiβjjρ(JB)]  }

. ()

In this paper, we continue to research the problems mentioned above. For anM-matrix B, we establish some new inequalities on the bounds forτ(B). Finally, some examples are given to illustrate our results.

For convenience, we employ the following notations throughout. LetA= (aij) be ann×n matrix. Fori,j,k∈N,i=j, denote

rji= | aji| |ajj|–_k₌_j_,_i|ajk|

, ri=max j=i {rji}, mji=

|aji|+_k₌_j_,_i|ajk|ri

|ajj| , hi=maxj=i

_|

aji| |ajj|mji–

k=j,i|ajk|mki

,

uji=

|aji|+_k₌_j_,_i|ajk|mkihi

|ajj| , ui=maxj=i {uij}. 2 Main results

In this section, we present our main results. Firstly, we give some notations and lemmas. LetA≥ andD=diag(aii). DenoteC=A–D,JA=D– C,D=diag(dii), where

dii=

, ifaii= , aii, ifaii= .

By the deﬁnition ofJA, we obtain

ρ(J_AT) =ρ

D–_ CT =ρCD–_ =ρD–_ CD–_ D =ρ

D–_ C =ρ(JA).

Lemma [] Let A∈Cn×n,and let x,x, . . . ,xn be positive real numbers.Then all the eigenvalues of A lie in the region

i

z∈C:|z–aii| ≤xi

j=i  xj

|aji|,i∈N

.

Lemma [] Let A∈Cn×n,and let x,x, . . . ,xnbe positive real numbers.Then all the eigenvalues of A lie in the region

j=i

z∈C:|z–aii||z–ajj| ≤

xi

k=i  xk

|aki|

xj

l=j  xl

|alj|

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Lemma [] Let A,B∈Rn×n_,_{and let X}_,_Y_∈_Rn×n_{be diagonal matrices}_._Then X(A◦B)Y= (XAY)◦B= (XA)◦(BY) = (AY)◦(XB) =A◦(XBY).

Lemma [] Let A= (aij)∈Mn.Then there exists a positive diagonal matrix X such that X–_{AX is a strictly diagonally dominant M-matrix}

Lemma [] Let A= (aij)∈Rn×n_{be a strictly diagonally dominant matrix and let A}–₌

(αij).Then for all i∈N,

αij≤ujiαjj, j∈N,j=i.

Theorem  Let A= (aij)≥,B= (bij)∈Mn,and let B–_{= (}_β_ij)._Then

ρA◦B– ≤max

≤i≤n

aii+uiρ(JA)dii βii

. ()

Proof It is evident that the result holds with equality forn= . We next assume thatn≥.

(i) First, we assume thatAandBare irreducible matrices. SinceBis anM-matrix, by Lemma , there exists a positive diagonal matrixX, such thatX–_BX_{is a strictly row}

di-agonally dominantM-matrix, and

ρA◦B– =ρX–A◦B– X =ρA◦X–BX – .

Hence, for convenience and without loss of generality, we assume thatBis a strictly diag-onally dominant matrix.

On the other hand, since A is irreducible and so isJ_AT. Then there exists a positive vectorx= (xi) such thatJATx=ρ(J_AT)x=ρ(J_A)x, thus, we obtain_j₌_ia_jix_j=ρ(J_A)d_iix_i. LetA= (a˜ij) =XAX–in whichXis the positive matrixX=diag(x,x, . . . ,xn). Then, we have

A= (a˜ij) =XAX–=

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

a ax_x · · · anx_xn

ax

x a · · · anx

xn ..

. ... . .. ... anxn

x

anxn

x · · · ann

⎞ ⎟ ⎟ ⎟ ⎟ ⎠.

From Lemma , we have

A◦B–=XAX– ◦B–=XA◦B– X–.

Thus, we obtainρ(A◦B–_{) =}_ρ₍_A_◦_B–_{). Let}_λ₌_ρ₍_A_◦_B–_{), so that}_λ_≥_a

iiβii,∀i∈N. By Lemma , there existsi∈N, such that

|λ–aiiβii| ≤ui

t=i  ut˜

atiβti≤ui

t=i  ut˜

atiutiβii

≤ui

t=i ˜

atiβii=uiβii

t=i atixt

xi

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Therefore,

λ≤aiiβii+uiρ(JA)diiβii=

aii+uiρ(JA)dii βii,

i.e.,

ρA◦B– ≤aii+uiρ(JA)dii βii ≤max

≤i≤n

aii+uiρ(JA)dii βii

.

(ii) Now, assume that one ofAandBis reducible. It is well known that a matrix inZnis a nonsingularM-matrix if and only if all its leading principal minors are positive (see []). If we denote byT= (tij) then×npermutation matrix witht=t=· · ·=tn–,n=tn= –,

the remainingtijzero, then bothA–εTandB+εTare irreducible matrices for any chosen positive real numberε, suﬃciently small such that all the leading principal minors ofB+εT are positive. Now, we substituteA–εTandB+εTforAandB, respectively, in the previous case, and then lettingε→, the result follows by continuity.

Theorem  Let B= (bij)∈Mnand B–= (βij).Then

τ(B)≥ 

max≤i≤n{( +ui(n– ))βii}. ()

Proof Let all entries ofAin () be . Thenaii=  (∀i∈N),ρ(JA) =n– . Therefore, by (), we have

τ(B) = 

ρ(B–₎≥



max__≤_i_≤n{( +ui(n– ))βii} .

The proof is completed.

Theorem  Let A= (aij)≥,B= (bij)∈Mn,and let B–_{= (}_β_ij)._Then

ρA◦B– ≤ 

maxi=j {aiiβii+ajjβjj+ ij}, () where ij= [(aiiβii–ajjβjj)+ uiujρ(JA)diidjjβiiβjj]

 _.

Proof It is evident that the result holds with equality forn= .

We next assume thatn≥. For convenience and without loss of generality, we assume thatBis a strictly row diagonally dominant matrix.

(i) First, we assume thatAandBare irreducible matrices. SinceAis irreducible and so is

JAT. Then there exists a positive vectory= (yi) such thatJ_ATy=ρ(J_AT)y=ρ(JA)y, thus, we obtain

k=i

akiyk=ρ(JA)diiyi,

k=j

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LetA= (ˆaij) =YAY–_{in which}_Y_{is the positive matrix}_Y₌_diag₍_y

,y, . . . ,yn). Then, we

have

A= (aˆij) =YAY–=

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

a ay_y · · · any_yn

ay

y a · · · any yn .. . ... . .. ... anyn y anyn

y · · · ann

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ .

From Lemma , we get

A◦B–=YAY– ◦B–=YA◦B– Y–.

Thus, we obtainρ(A◦B–) =ρ(A◦B–). Letλ=ρ(A◦B–), so thatλ≥aiiβii(∀i∈N). By Lemma , there existi,j∈N,i=j, such that

|λ–aiiβii||λ–ajjβjj| ≤

ui

k=i  uk

ˆ akiβki

uj

k=j  uk

ˆ akjβkj

.

Note that

ui

k=i  ukˆ

akiβki≤ui

k=i  ukˆ

akiukiβii≤uiβii

k=i ˆ

aki=uiβiiρ(JA)dii,

uj

k=j  uk

ˆ

akjβkj≤uj

k=j  uk

ˆ

akjukjβjj≤ujβjj

k=j ˆ

akj=ujβjjρ(JA)djj.

Hence, we obtain

λ≤ 

(aiiβii+ajjβjj+ ij),

i.e.,

ρA◦B– ≤ 

(aiiβii+ajjβjj+ ij)≤ 

maxi=j {aiiβii+ajjβjj+ ij},

where ij= [(aiiβii–ajjβjj)+ uiujρ(JA)diidjjβiiβjj]  .

(ii) Now, assume that one ofAandBis reducible. We substituteA–εTandB+εTfor AandB, respectively, in the previous case (as in the proof of Theorem ), and then letting

ε→, the result follows by continuity.

Theorem  Let B= (bij)∈Mnand B–= (βij).Then

τ(B)≥ 

maxi=j{βii+βjj+ ij}

, ()

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Proof Let all entries ofAin () be . Then

aii=  (∀i∈N), ρ(JA) =n– , ij=

(βii–βjj)+ (n– )uiujβiiβjj

 _.

Therefore, by (), we have

τ(B) = 

ρ(B–₎≥



maxi=j{βii+βjj+ ij} .

The proof is completed.

Remark  We next give a simple comparison between () and (), () and (), respectively. For convenience and without loss of generality, we assume that, fori,j∈N,i=j,

ajjβjj+ujdjjβjjρ(JA)≤aiiβii+uidiiβiiρ(JA),

i.e.,

ujdjjβjjρ(JA)≤aiiβii–ajjβjj+uidiiβiiρ(JA).

Hence,

ij=

(aiiβii–ajjβjj)+ uiujρ(JA)diidjjβiiβjj

 

≤(aiiβii–ajjβjj)+ uiρ(JA)diiβii

aiiβii–ajjβjj+uidiiβiiρ(JA)  

=aiiβii–ajjβjj+ uidiiβiiρ(JA).

Further, we obtain

aiiβii+ajjβjj+ ij≤aiiβii+ uidiiβiiρ(JA),

i.e.,

ρA◦B– ≤ 

maxi=j {aiiβii+ajjβjj+ ij} ≤max≤i≤n

aii+uiρ(JA)dii βii

.

So, the bound in () is better than the bound in (). Similarly, we can prove that the bound in () is better than the bound in ().

3 Numerical examples

In this section, we present numerical examples to illustrate the advantages of our derived results.

Example  Let

B=

⎛ ⎜ ⎝

. –. –. –.  –. –. –. .

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It is easy to see thatBis anM-matrix. By calculations with Matlab ., we have

τ(B)≥. (by ()), τ(B)≥. (by ()),

τ(B)≥. (by ()), τ(B)≥. (by ()),

τ(B)≥. (by ()),

respectively. In fact,τ(B) = .. It is obvious that the bound in () is the best result.

Example  Let

B=

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

 –. –. –. –. –.  –. –. –. –. –.  –. –. –. –. –.  –. –. –. –. –. 

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

.

It is easy to see thatBis anM-matrix. By calculations with Matlab ., we have

τ(B)≥. (by ()), τ(B)≥. (by ()),

τ(B)≥. (by ()), τ(B)≥. (by ()),

τ(B)≥. (by ()),

respectively. In fact,τ(B) = .. It is obvious that the bound in () is the best result.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally to this work. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The authors are very indebted to the referees for their valuable comments and corrections, which improved the original manuscript of this paper. This work was supported by National Natural Science Foundations of China (11361074, 71161020), Applied Basic Research Programs of Science and Technology Department of Yunnan Province (2013FD002), and IRTSTYN, Research Foundation of Guizhou Minzu University (15XRY004).

Received: 21 April 2015 Accepted: 1 June 2015

References

1. Berman, A, Plemmons, RJ: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York (1979) 2. Fiedler, M, Markham, T: An inequality for the Hadamard product of anM-matrix and inverseM-matrix. Linear Algebra

Appl.101, 1-8 (1988)

3. Horn, RA, Johnson, CR: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991) 4. Corduneanu, C: Principles of Diﬀerential and Integral Equations. Chelsea, New York (1988)

5. Shivakumar, PN, Williams, JJ, Ye, Q, Marinov, CA: On two-sided bounds related to weakly diagonally dominant M-matrices with application to digital circuit dynamics. SIAM J. Matrix Anal. Appl.17, 298-312 (1996) 6. Walter, W: Diﬀerential and Integral Inequalities. Springer, Berlin (1970)

7. Tian, GX, Huang, TZ: Inequalities for the minimum eigenvalue ofM-matrices. Electron. J. Linear Algebra20, 291-302 (2010)

8. Li, CQ, Li, YT, Zhao, RJ: New inequalities for the minimum eigenvalue ofM-matrices. Linear Multilinear Algebra61(9), 1267-1279 (2013)

9. Varga, RS: Geršcgorin and His Circles. Springer, Berlin (2004)