Some inequalities for the minimum eigenvalue of the Hadamard product of an M matrix and its inverse

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R E S E A R C H

Open Access

Some inequalities for the minimum

eigenvalue of the Hadamard product of an

M

-matrix and its inverse

Guanghui Cheng

*

_{, Qin Tan and Zhuande Wang}

*_{Correspondence:}

[email protected] School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, P.R. China

Abstract

In this paper, some new inequalities for the minimum eigenvalue of the Hadamard product of anM-matrix and its inverse are given. These inequalities are sharper than the well-known results. A simple example is shown.

AMS Subject Classiﬁcation: 15A18; 15A42

Keywords: Hadamard product;M-matrix; inverseM-matrix; strictly diagonally dominant matrix; eigenvalue

1 Introduction

A matrixA= (aij)∈Rn×nis called a nonnegative matrix ifaij≥. A matrixA∈Rn×nis

called a nonsingularM-matrix [] if there existB≥ ands>  such that

A=sIn–B and s>ρ(B),

whereρ(B) is a spectral radius of the nonnegative matrixB,Inis then×nidentity matrix.

Denote byMnthe set of alln×nnonsingularM-matrices. The matrices inM–n :={A–:

A∈Mn}are called inverseM-matrices. Let us denote

τ(A) =minReλ:λ∈σ(A),

andσ(A) denotes the spectrum ofA. It is known that []

τ(A) = 

ρA–

is a positive real eigenvalue ofA∈Mnand the corresponding eigenvector is nonnegative.

Indeed

τ(A) =s–ρ(B),

ifA=sIn–B, wheres>ρ(B),B≥.

For any twon×nmatricesA= (aij) andB= (bij), the Hadamard product ofAandBis

A◦B= (aijbij). IfA,B∈Mn, thenA◦B–is also anM-matrix [].

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A matrixAis irreducible if there does not exist a permutation matrixPsuch that

PAPT=

A, A,

 A,

,

whereA,andA,are square matrices.

For convenience, the set{, , . . . ,n}is denoted byN, wheren(≥) is any positive integer. LetA= (aij)∈Rn×nbe a strictly diagonally dominant by row, denote

Ri=

k=i

|aik|, di=

Ri

|aii|

, ∀i∈N;

sji=

|aji|+ k=j,i|ajk|dk

|ajj|

, j=i,∀j∈N; si=max

j=i {sij}, ∀i∈N;

rji=

|aji|

|ajj|– k=j,i|ajk|

, j=i,∀j∈N; ri=max

j=i {rji}, ∀i∈N;

mji=

|aji|+ k=j,i|ajk|ri

|ajj|

, j=i,∀j∈N; mi=max

j=i {mij}, ∀i∈N;

uji=

|aji|+ k=j,i|ajk|mki

|ajj|

, j=i,∀j∈N; ui=max

j=i {uij}, ∀i∈N.

Recently, some lower bounds for the minimum eigenvalue of the Hadamard product of anM-matrix and an inverseM-matrix have been proposed. LetA∈Mn, for

exam-ple,τ(A◦A–₎_≤_{ has been proven by Fiedler}_{et al.}_{in []. Subsequently,}_τ₍_A_◦_A–_{) >} 

n

was given by Fiedler and Markham in [], and they conjectured thatτ(A◦A–_{) >}

n. Song

[], Yong [] and Chen [] have independently proven this conjecture. In [], Liet al. im-proved the conjectureτ(A◦A–)≥_nwhenA–is a doubly stochastic matrix and gave the following result:

τA◦A–≥min i

aii–siRi

 + _j₌_isji

.

In [], Liet al.gave the following result:

aii–miRi

 + _j₌_imji

.

Furthermore, ifa=a=· · ·=ann, they have obtained

min i

aii–miRi

 + _j₌_imji

≥min

i

aii–siRi

 + _j₌_isji

,

i.e., under this condition, the bound of [] is better than the one of [].

In this paper, our motives are to improve the lower bounds for the minimum eigenvalue

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2 Some preliminaries and notations

In this section, we give some notations and lemmas which mainly focus on some in-equalities for the entries of the inverse M-matrix and the strictly diagonally dominant matrix.

Lemma .[] Let A∈Rn×n_{be a strictly diagonally dominant matrix by row}_,_i.e._,

|aii|>

j=i

|aij|, ∀i∈N.

If A–_{= (}_b

ij),then

|bji| ≤ k=j

|ajk|

|ajj| |

bii|, j=i,∀j∈N.

Lemma . Let A∈Rn×n_{be a strictly diagonally dominant M-matrix by row}_._{If A}–_{= (}_b

ij),

then

bji≤

ajj

bii≤ujbii, j=i,∀i∈N.

Proof Firstly, we considerA∈Rn×n_{is a strictly diagonally dominant}_M_{-matrix by row. For}

i∈N, let

ri(ε) =max j=i

_| aji|+ε

ajj– k=j,i|ajk|

and

mji(ε) =

ri(ε)( k=j,i|ajk|+ε) +|aji|

ajj

, j=i.

SinceAis strictly diagonally dominant, thenrji<  andmji< . Therefore, there existsε> 

such that  <ri(ε) <  and  <mji(ε) < . Let us deﬁne one positive diagonal matrix

Mi(ε) =diag

mi(ε), . . . ,mi–,i(ε), ,mi+,i(ε), . . . ,mni(ε)

.

Similarly to the proofs of Theorem . and Theorem . in [], we can prove that the matrixAMi(ε) is also a strictly diagonally dominantM-matrix by row for anyi∈N.

Fur-thermore, by Lemma ., we can obtain the following result:

m–_ji (ε)bji≤

|aji|+ k=j,i|ajk|mki(ε)

mji(ε)ajj

bii, j=i,j∈N,

i.e.,

bji≤

|aji|+ k=j,i|ajk|mki(ε)

ajj

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Letε–→+_{to get}

bji≤

ajj

bii≤ujbii, j=i,j∈N.

This proof is completed.

Lemma . Let A= (aij)∈Mnbe a strictly diagonally dominant matrix by row and A–=

(bij),then we have

 aii≤

bii≤

 aii– j=i|aij|uji

, ∀i∈N.

Proof LetB=A–_{. Since}_A_{is an}_M_{-matrix, then}_B_≥_{. By}_AB₌_BA₌_I

n, we have

 =

n

j=

aijbji=aiibii–

j=i

|aij|bji, ∀i∈N.

Hence

≤aiibii, ∀i∈N,

or equivalently,

 aii≤

bii, ∀i∈N.

Furthermore, by Lemma ., we get

 =aiibii–

j=i

|aij|bji≥

aii–

j=i

|aij|uji

bii, ∀i∈N,

i.e.,

bii≤

 aii– j=i|aij|uji

, ∀i∈N.

Thus the proof is completed.

Lemma .[] Let A∈Cn×n _{and x}

,x, . . . ,xn be positive real numbers.Then all the

eigenvalues of A lie in the region

n



z∈C:|z–aii| ≤xi

j=i

 xj|

aji|

.

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3 Main results

In this section, we give two new lower bounds forτ(A◦A–) which improve the ones in [] and [].

Lemma . If A∈Mnand A–= (bij)is a doubly stochastic matrix,then

bii≥

  + _j₌_iuji

, ∀i∈N.

Proof This proof is similar to the ones of Lemma . in [] and Theorem . in [].

Theorem . Let A∈Mnand A–= (bij)be a doubly stochastic matrix.Then

aii–uiRi

 + _j₌_iuji

.

Proof Firstly, we assume thatAis irreducible. By Lemma ., we have

aii=

j=i

|aij|+  =

j=i

|aji|+  and aii> , i∈N.

Denote

uj=max

i=j {uji}=max

_|_a

ji|+ k=j,i|ajk|mki

ajj

, j∈N.

SinceAis an irreducible matrix, we know that  <uj≤. So, by Lemma ., there exists

i∈Nsuch that

|λ–aiibii| ≤ui

j=i  uj|

ajibji|,

or equivalently,

|λ| ≥aiibii–ui

j=i  uj|

ajibji|

≥aiibii–ui

j=i  uj|

aji|ujbii (by Lemma .)

≥

aii–ui

j=i |aji|

bii

= (aii–uiRi)bii

≥ aii–uiRi  + _j₌_i_uji

(by Lemma .)

≥min i

aii–uiRi

 + _j₌_iuji

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Secondly, ifAis reducible, without loss of generality, we may assume that Ahas the following block upper triangular form:

A= ⎡ ⎢ ⎢ ⎢ ⎣

A A · · · AK

 A · · · AK

  · · · ·    AKK

⎤ ⎥ ⎥ ⎥ ⎦,

whereAii∈Mni is an irreducible diagonal block matrix,i= , , . . . ,K. Obviously,τ(A◦

A–) =miniτ(Aii◦A–ii). Thus the reducible case is converted into the irreducible case. This

proof is completed.

Theorem . If A= (aij)∈Mnis a strictly diagonally dominant by row,then

min i

aii–uiRi

 + _j₌_iuji

≥min

i

aii–siRi

 + _j₌_isji

.

Proof SinceAis strictly diagonally dominant by row, for anyj=i, we have

dj–mji=

|aji|+ k=j,i|ajk|

ajj

–|aji|+ k=j,i|ajk|ri ajj

= ( –ri) k=j,i|ajk| ajj

≥,

or equivalently,

dj≥mji, j= i,∀j∈N. ()

So, we can obtain

uji=

ajj

≤|aji|+ k=j,i|ajk|dk

ajj

=sji, j=i,∀j∈N, ()

and

ui≤si, ∀i∈N.

Therefore, it is easy to obtain that

aii–uiRi

 + _j₌_iuji

≥ aii–siRi

 + _j₌_isji

, ∀i∈N.

Obviously, we have the desired result

min i

aii–uiRi

 + _j₌_iuji

≥min

i

aii–siRi

 + _j₌_isji

.

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Theorem . If A= (aij)∈Mnis strictly diagonally dominant by row,then

min i

aii–uiRi

 + _j₌_iuji

≥min

i

aii–miRi

 + _j₌_imji

.

Proof SinceAis strictly diagonally dominant by row, for anyj=i, we have

ri–mji=ri–

ajj

= ri–|aji|– k=j,i|ajk| ajj

= ri(ajj– k=j,i|ajk|) –|aji| ajj

= ajj– k=j,i|ajk| ajj

ri–

|aji|

ajj– k=j,i|ajk|

≥,

i.e.,

ri≥mji, j=i,∀j∈N. ()

So, we can obtain

uji=

ajj

≤|aji|+ k=j,i|ajk|ri

ajj

=mji, j=i,∀j∈N, ()

and

ui≤mi, ∀i∈N.

Therefore, it is easy to obtain that

aii–uiRi

 + _j₌_iuji

≥ aii–miRi

 + _j₌_imji

, ∀i∈N.

Obviously, we have the desired result

min i

aii–uiRi

 + _j₌_iuji

≥min

i

aii–miRi

 + _j₌_imji

.

Remark . According to inequalities () and (), it is easy to know that

bji≤

ajj

bii≤

|aji|+ k=j,i|ajk|dk

ajj

bii, ∀i∈N.

and

bji≤

ajj

bii≤

ajj

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That is to say, the result of Lemma . is sharper than the ones of Theorem . in [] and Lemma . in []. Moreover, the results of Theorem . and Theorem . are sharper than the ones of Theorem . in [] and Theorem . in [], respectively.

Theorem . If A∈Mnis strictly diagonally dominant by row,then

 – 

aii

j=i

|aji|uji

.

Proof This proof is similar to the one of Theorem . in [].

Remark . According to inequalities () and (), we get

 –  aii

j=i

|aji|uji≥ –

 aii

j=i

|aji|sji,

and

 –  aii

j=i

|aji|uji≥ –

 aii

j=i

|aji|mji.

That is to say, the bound of Theorem . is sharper than the ones of Theorem . in [] and Theorem . in [], respectively.

Remark . Using the above similar ideas, we can obtain similar inequalities of the strictly diagonallyM-matrix by column.

4 Example

For convenience, we consider theM-matrixAis the same as the matrix of []. Deﬁne the M-matrixAas follows:

A= ⎡ ⎢ ⎢ ⎢ ⎣  – – – –  – –  –  – – – –  ⎤ ⎥ ⎥ ⎥ ⎦.

. Estimate the upper bounds for entries ofA–_{= (}_b

ij). Firstly, by Lemma .() in [],

we have

A–≤ ⎡ ⎢ ⎢ ⎢ ⎣  . . . .  . . . .  . . . .  ⎤ ⎥ ⎥ ⎥ ⎦◦ ⎡ ⎢ ⎢ ⎢ ⎣

b b b b

⎤ ⎥ ⎥ ⎥ ⎦.

By Lemma ., we have

A–≤ ⎡ ⎢ ⎢ ⎢ ⎣  . . . .  . . . .  . . . .  ⎤ ⎥ ⎥ ⎥ ⎦◦ ⎡ ⎢ ⎢ ⎢ ⎣

b b b b

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By Lemma . and Theorem . in [], we get

.≤b≤., .≤b≤.,

.≤b≤., .≤b≤..

By Lemma . and Lemma ., we get

.≤b≤., .≤b≤.,

.≤b≤., .≤b≤..

. Lower bounds forτ(A◦A–). By Theorem . in [], we obtain

. =τA◦A–≥..

By Theorem ., we obtain

. =τA◦A–≥..

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the ﬁnal manuscript.

Acknowledgements

This research is supported by National Natural Science Foundations of China (No. 11101069).

Received: 31 July 2012 Accepted: 24 January 2013 Published: 21 February 2013 References

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doi:10.1186/1029-242X-2013-65

Cite this article as:Cheng et al.:Some inequalities for the minimum eigenvalue of the Hadamard product of an