Properties for the Perron complement of three known subclasses of H matrices

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R E S E A R C H

Open Access

Properties for the Perron complement of

three known subclasses of

H

-matrices

Leilei Wang, Jianzhou Liu

*

_{and Shan Chu}

*_{Correspondence: [email protected]}

Department of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, P.R. China

Abstract

In this paper, we analyze the diagonally dominant degree for the Perron complement upon several diagonally dominant cases by using the entries and spectral radius of the original matrix. At the same time, we obtain closure properties for the Perron complement of several diagonally dominant matrices.

MSC: Primary 15A47; 15A48; secondary 65U05; 65J10

Keywords: diagonally dominant matrix;H-matrix; the Perron complement; nonnegative irreducible matrix; spectral radius

1 Introduction

The Perron complement plays an important role in statistics, matrix theory, control the-ory, and so on (see [–]). It was said in [, ] that the Perron vector (or the stationary dis-tribution vector) for a corresponding transition matrix of a certain Markov chain provides the long term probabilities of the chain being in a particular state. Meyer [] introduced the concept of the Perron complement, obtained the Perron complement of a nonnegative irreducible matrix is nonnegative irreducible, and ﬁrst used the closure property of a non-negative irreducible matrix to construct a divided and conquer algorithm to compute the Perron vector for a Markov chain. Since then, a lot of work has been done on this topic. The authors in [–] showed closure properties of the Perron complement for some spe-cial matrices. When the Perron complement is primitive, we can design a related iterative algorithm to compute the Perron vector (see []). So far as we know, if the given matrix has a sharper diagonally dominant degree, then the designed iterative algorithms has faster convergent rate than the ordinary ones (see []). At the same time, if a given matrix has a sharper diagonally dominant degree, then we may discuss more properties about general-ized nonlinear diagonal dominance in []. Motivated by the useful applications, we will study the diagonal dominant degree for the Perron complement of several cases based on the nonnegative and irreducible nature, which may belong to inherent properties of the Perron complement. Meanwhile, some closure properties for the Perron complement of three known subclasses ofH-matrices are provided by the entries and spectral radius of the original matrix.

LetCn×n(Rn×n) denote the set of alln×ncomplex (real) matrices,N={, , . . . ,n}, and G={(i,j) :i,j∈N,i=j}. ForA= (aij)∈Cn×n, denote

Ri(A) =

j∈N,j=i

|aij|, Si(A) =

j∈N,j=i

|aji|, i∈N.

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Choose

Nr(A) =

i:|aii|>Ri(A),i∈N

, Nc(A) =

j:|ajj|>Sj(A),j∈N

.

The comparison matrixμ(A) = (μij) ofA= (aij)∈Cn×n, deﬁned by

μij=

|aii|, i=j,

–|aij|, i=j.

IfA=μ(A) and the eigenvalues ofAhave positive real parts, then we callAa (nonsingular)

M-matrix. We say thatAis anH-matrix ifμ(A) is anM-matrix. For details and numerous conditions ofM-matrices, please refer to []. We denote byHnandMnthe sets of alln×n

H- andM-matrices, respectively.

Recall thatA= (aij)∈Cn×nis (row) diagonally dominant and denoted byA∈Dnif

|aii| ≥Ri(A) (.)

for alli∈Nholds.Ais said to be a strictly diagonally dominant matrix and denoted by

A∈SDnif the representative inequality (.) is strict.

A= (aij)∈Cn×nis doubly diagonally dominant and denoted byA∈DDnif

|aii||ajj| ≥Ri(A)Rj(A) (.)

for all (i,j)∈Gvalid. Moreover,Ais said to be a strictly doubly diagonally dominant matrix and denoted byA∈SDDnif the representative inequality (.) is strict for all (i,j)∈G. If

A∈SDDn, then there exists at most one indexi∈Nsuch that

|aii| ≤Ri(A). (.)

A= (aij)∈Cn×nisγ-diagonally dominant and denoted byA∈D

γ

nif there existsγ∈[, ]

such that

|aii| ≥γRi(A) + ( –γ)Si(A), for alli∈N. (.)

If all the inequalities in (.) are strict, thenAis called strictγ-diagonally dominant.

A= (aij)∈Cn×n is productγ-diagonally dominant and denoted by A∈PD

γ

n if there

existsγ ∈[, ] such that

|aii| ≥

Ri(A)

γ Si(A)

–γ

, for alli∈N. (.)

If all the inequalities in (.) are strict, thenAis said to be a strictly productγ-diagonally dominant matrix.

ForA∈Cn×n_{, nonempty index sets}_α_,_β_⊆_N_{, we denote by}_|_α_|_{the cardinality of}_α_{. Let}

A(α,β) denote the sub-matrix ofAlying in the rows indexed byαand the columns indexed byβ.A(α,α) is abbreviated toA(α). IfA(α) is nonsingular, then the Schur complement of

A(α) inAis given by

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LetAbe a nonnegative irreducible matrix of ordernwith spectral radiusρ(A),∅ =α, andβ=N–α. Then the Perron complement ofAwith respect toA(α), which is denoted byP(A/A(α)) or simplyP(A/α), is deﬁned as

A(β) +A(β,α)ρ(A)I–A(α)–A(α,β). (.)

In addition, the extended Perron complementPt(A/A(α)) or simplyPt(A/α) attis deﬁned

as

A(β) +A(β,α)tI–A(α)–A(α,β). (.)

2 Properties for the Perron complement of diagonally dominant matrices

In this section, we oﬀer some properties for the Perron complement of diagonally domi-nant matrices by using the entries and spectral radius of the original matrix.

Lemma (see []) If A is an H-matrix,then[μ(A)]–_{≥ |}_A_|–_.

Lemma (see []) If A= (aij)∈SDn,or A∈SDDn,thenμ(A)∈Mn,i.e.,A∈Hn.

Proposition  Let A= (aij)∈Rn×nis nonnegative irreducible with spectral radiusρ(A),

α={i,i, . . . ,ik} ⊆Nr(A),β=N–α={j,j, . . . ,jl},|α|<n,and denote

Bjt≡

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

x –|ajti| · · · –|ajtik|

–l_u_=|aiju|

..

. μ[ρ(A)I–A(α)] –l_u_=|aikju|

⎞ ⎟ ⎟ ⎟ ⎟

⎠, x> .

Then forρ(A)≥|aii|(i∈α)and any jt∈β,Bjtis an M-matrix of order k+ anddetBjt>  if

x>

k

ω= |ajtiω|

Riω(A)

ρ(A) –|aiωiω|

. (.)

Proof DenoteBjt≡B≡(bpq). Equation (.) means that there existsε>  such that

x>

k

ω= |ajtiω|

Riω(A)

ρ(A) –|aiωiω|

+ε

. (.)

Byρ(A)≥|aii|(i∈α) andα⊆Nr(A), we have ≤ Riω(A)

ρ(A)–|aiωiω|<  (≤ω≤k). Choose a

positive matrixD=diag(d,d, . . . ,dk+) andK=BD= (ksv), where

dv=

, v= ;

R_iv_–(A)

ρ(A)–|a_iv_–_iv_–|+ε, ≤v≤k+ .

(i) Fors= , (.) follows from

|kss|–

v=s

|ksv|=|k|–

k+

v=

|kv|=x– k+

v= |ajtiv–|

Riω(A)

ρ(A) –|aiωiω|

+ε

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(ii) Fors= , , . . . ,k+ , we obtain

|kss|–

v=s

|ksv|

=ρ(A) –|ais–is–|

Ris–(A) ρ(A) –|ais–is–|

+ε

–

l

u=

|ais–ju|–

k+

v=,v=s

|ais–iv–|

Riv–(A) ρ(A) –|aiv–iv–|

+ε

=Ris–(A) –

l

u=

|ais–ju|–

k

v=,v=s– |ais–iv|

Riv(A)

ρ(A) –|aiviv|

+ε

ρ(A) –|ais–is–|–

k+

v=,v=s

|ais–iv–|

> .

Thus,K∈SDk+. Further,Bjt∈Hk+. Besides, observing thatBjt =μ(Bjt), we have Bjt ∈

Mk+anddetBjt> .

Proposition  Let A= (aij)∈Rn×nbe nonnegative irreducible with spectral radiusρ(A),

α={i,i, . . . ,ik} ⊆N,β=N–α={j,j, . . . ,jl},|α|<n,P(A/α) = (ats),andρ(A)≥|aii|

(i∈α).

(i)Ifα⊆Nr(A),then for all≤t≤l,

l

s=,s=t

ajtjs+ (ajti, . . . ,ajtik)

ρ(A)I–A(α)–

⎛ ⎜ ⎜ ⎝

aijs

.. .

aikjs

⎞ ⎟ ⎟ ⎠

+

(ajti, . . . ,ajtik)

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠

≤Rjt(A) –ωjt. (.)

(ii)Ifα⊆Nc(A),then for all≤t≤l,

l

s=,s=t

ajsjt+ (ajsi, . . . ,ajsik)

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠

+

(ajti, . . . ,ajtik)

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠

≤Sjt(A) –ω

T

jt. (.)

Here

ωjt=

k

u= |ajtiu|

ρ(A) –|aiuiu|–Riu(A)

ρ(A) –|aiuiu|

, ωT_j_t =

k

u= |aiujt|

ρ(A) –|aiuiu|–Siu(A)

ρ(A) –|aiuiu|

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Proof Byα⊆Nr(A), we have A(α)∈SDk. Further,ρ(A)≥|aii| (i∈α) yields (ρ(A)I–

A(α))∈SDk. By Lemma  and Lemma , we have

μρ(A)I–A(α)–≥ρ(A)I–A(α)–.

Thus, by the deﬁnition of the Perron complement, we obtain

(ajti, . . . ,ajtik)

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠

+

l

s=,s=t

⎛ ⎜ ⎜ ⎝

aijs

.. .

aikjs

⎞ ⎟ ⎟ ⎠

≤

l

s=,s=t

|ajtjs|+

l

s=

(ajti, . . . ,ajtik)

⎛ ⎜ ⎜ ⎝

aijs

.. .

aikjs

⎞ ⎟ ⎟ ⎠

≤

l

s=,s=t

|ajtjs|+

l

s=

|ajti|, . . . ,|ajtik| μ

⎛ ⎜ ⎜ ⎝

|aijs|

.. .

|aikjs|

⎞ ⎟ ⎟ ⎠

=Rjt(A) –

k

u= |ajtiu|+

|ajti|, . . . ,|ajtik| μ

⎛ ⎜ ⎜ ⎝

l s=|aijs|

.. .

l s=|aikjs|

⎞ ⎟ ⎟ ⎠

=Rjt(A) –ωjt–



det{μ[ρ(A)I–A(α)]}

×det

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

k

u=|ajtiu|–ωjt –|ajti| · · · –|ajtik|

–l_s_=|aijs|

..

. μ[ρ(A)I–A(α)] –l_s_=|aikjs|

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

≡Rjt(A) –ωjt–

detB

det{μ[ρ(A)I–A(α)]}.

On the basis of Proposition , we havedetB> , which implies (.). Moreover, (.) can

be proved with a similar method to the above techniques.

Theorem  If A= (aij)∈Rn×nis nonnegative irreducible with spectral radiusρ(A),α=

{i,i, . . . ,ik} ⊆Nr(A),β=N–α={j,j, . . . ,jl},|α|<n,P(A/α) = (ats),then forρ(A)≥|aii|

(i∈α)and≤t≤l,

a_tt–Rt

P(A/α) ≥ |ajtjt|–Rjt(A) +ωjt≥ |ajtjt|–Rjt(A) (.) and

a_tt+Rt

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Proof Byα⊆Nr(A) and Proposition , we have

a_tt–Rt

P(A/α) =a_tt–

l

s=,s=t

a_ts

=

ajtjt+ (ajti, . . . ,ajtik)

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠

–

l

s=,s=t

⎛ ⎜ ⎜ ⎝

aijs

.. .

aikjs

⎞ ⎟ ⎟ ⎠

≥ |ajtjt|–

(ajti, . . . ,ajtik)

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠

–

l

s=,s=t

⎛ ⎜ ⎜ ⎝

aijs

.. .

aikjs

⎞ ⎟ ⎟ ⎠ ≥ |ajtjt|–

Rjt(A) –ωjt ,

which implies (.).

Similarly, we can also verify (.) immediately.

WhenA∈SDn, (.) and Theorem . of [] show the following fact.

Corollary  If A= (aij)∈Rn×nis nonnegative irreducible and strictly diagonally dominant

with spectral radiusρ(A),∅ =α⊆N,|α|<n,β=N–α,then forρ(A)≥|aii|(i∈α),

P(A/α) =A(β) +A(β,α)ρ(A)I–A(α)–A(α,β)

is nonnegative irreducible and strictly diagonally dominant.

3 Properties for the Perron complement of strictly

γ

- and product

γ

-diagonally dominant matrices

In this section, we obtain several properties for the Perron complement of strictlyγ- and productγ-diagonally dominant matrices through using the entries and spectral radius of the original matrix.

Lemma (see []) If a>b,c>b,b> ,and≤r≤,then

arc–r≥(a–b)r(c–b)–r+b.

Theorem  If A= (aij)∈Rn×nis nonnegative irreducible and with spectral radiusρ(A),

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and P(A/α) = (a_ts),then forρ(A)≥|aii|(i∈α), ≤t≤l,and≤γ ≤,

a_tt–γRt

P(A/α) – ( –γ)St

P(A/α)

≥ |ajtjt|–γRjt(A) – ( –γ)Sjt(A) +γ ωjt+ ( –γ)ω

T jt

≥ |ajtjt|–γRjt(A) – ( –γ)Sjt(A)

and

a_tt+γRt

P(A/α) + ( –γ)St

P(A/α)

≤ |ajtjt|+γRjt(A) + ( –γ)Sjt(A) –γ ωjt– ( –γ)ωTjt

≤ |ajtjt|+γRjt(A) + ( –γ)Sjt(A).

Proof In light ofα⊆(Nr(A)∩Nc(A)) and Proposition , we have

a_tt–γRt

P(A/α) – ( –γ)St

P(A/α)

=a_tt–γ

l

s=,s=t

a_ts– ( –γ)

l

s=,s=t

a_st

=

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠ –γ l

s=,s=t

⎛ ⎜ ⎜ ⎝

aijs

.. .

aikjs

⎞ ⎟ ⎟ ⎠

– ( –γ)

l

s=,s=t

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠

≥ |ajtjt|–

(ajti, . . . ,ajtik)

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠ –γ l

s=,s=t

⎛ ⎜ ⎜ ⎝

aijs

.. .

aikjs

⎞ ⎟ ⎟ ⎠

– ( –γ)

l

s=,s=t

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠

=|ajtjt|–γ

(ajti, . . . ,ajtik)

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

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–γ

l

s=,s=t

⎛ ⎜ ⎜ ⎝

aijs

.. .

aikjs

⎞ ⎟ ⎟ ⎠

– ( –γ)

(ajti, . . . ,ajtik)

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠

– ( –γ)

l

s=,s=t

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠ ≥ |ajtjt|–γ

Rjt(A) –ωjt – ( –γ)

Sjt(A) –ω

T jt

=|ajtjt|–γRjt(A) – ( –γ)Sjt(A) +γ ωjt+ ( –γ)ω

T jt,

which yields the ﬁrst type of inequalities. Similarly, the rest of the inequalities of

Theo-rem  can be veriﬁed.

By Theorem  and Theorem . of [], we get the following corollary.

Corollary  If A= (aij)∈Rn×nis nonnegative irreducible and strictlyγ-diagonally

dom-inant with spectral radiusρ(A),Nr(A)∩Nc(A)=∅,α⊆Nr(A)∩Nc(A),|α|<n,β=N–α,

then forρ(A)≥|aii|(i∈α),

is nonnegative irreducible and strictlyγ-diagonally dominant,and so is Pt(A/α).

Theorem  If A= (aij)∈ Rn×n is nonnegative irreducible with spectral radius ρ(A),

Nr(A)∩Nc(A)=∅, α={i,i, . . . ,ik} ⊆Nr(A)∩Nc(A), |α|<n,β =N–α={j,j, . . . ,jl},

P(A/α) = (a_ts),then forρ(A)≥|aii|(i∈α), ≤t≤l,and≤γ ≤,

a_tt–Rγ_tP(A/α)S–_t γP(A/α)

≥ |ajtjt|–

Rjt(A) –ωjt

γ

Sjt(A) –ω

T jt

–γ

≥ |ajtjt|–R γ

jt(A)S

–γ

jt (A) and

a_tt+Rγ_tP(A/α)S–_t γP(A/α)

≤ |ajtjt|+

Rjt(A) –ωjt

γ

Sjt(A) –ω

T jt

–γ

≤ |ajtjt|+R γ

jt(A)S

–γ

jt (A). Proof By the deﬁnition of the Perron complement, we obtain

a_tt–Rt

P(A/α) γSt

P(A/α) –γ

=a_tt–

_l

s=,s=t

a_ts

γ _l

s=,s=t

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=

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠ – ⎡ ⎢ ⎢ ⎣ l

s=,s=t

⎛ ⎜ ⎜ ⎝

aijs

.. .

aikjs

⎞ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎦ γ × ⎡ ⎢ ⎢ ⎣ l

s=,s=t

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎦ –γ

≥ |ajtjt|–

(ajti, . . . ,ajtik)

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠ – ⎡ ⎢ ⎢ ⎣ l

s=,s=t

⎛ ⎜ ⎜ ⎝

aijs

.. .

aikjs

⎞ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎦ γ × ⎡ ⎢ ⎢ ⎣ l

s=,s=t

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎦ –γ . Denote h=

(ajti, . . . ,ajtik)

⎛ ⎜ ⎜ ⎝

aijt

.. .

aikjt

⎞ ⎟ ⎟ ⎠ .

In light of Lemma , we get

a_tt–Rt

P(A/α) γSt

P(A/α) –γ

≥ |ajtjt|–h–

Rjt(A) –ωjt–h

γ

Sjt(A) –ωTjt –h

–γ

≥ |ajtjt|–h–

Rjt(A) –ωjt γ

Sjt(A) –ω

T jt –h

=|ajtjt|–

Rjt(A) –ωjt

γ

Sjt(A) –ω

T jt

–γ

.

Hence, we can get the ﬁrst type of inequalities of Theorem . Similarly, we can immediately

verify the other one.

By Theorem  and Theorem . of [], we have the following corollary.

Corollary  If A= (aij)∈Rn×nis nonnegative irreducible and strictly productγ-diagonally

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β=N–α,then forρ(A)≥|aii|(i∈α),

is nonnegative irreducible and strictly productγ-diagonally dominant,and so is Pt(A/α).

Competing interests

The authors declare that they have no competing interests.

Authors? contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The author thanks the referee for the very helpful comments and suggestions. This work was supported in part by National Natural Science Foundation of China (91430213), National Natural Science Foundation of China (11471279), National Natural Science Foundation for Youths of China (11401505), the Key Project of Hunan Provincial Natural Science Foundation of China (2015JJ2134), the Key Project of Hunan Provincial Education Department of China (12A137) and the Hunan Provincial Innovation Foundation for Postgraduate (CX2014B254).

Received: 2 September 2014 Accepted: 13 December 2014

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