Symmetric stochastic inverse eigenvalue problem

17  Download (0)

Full text

(1)

R E S E A R C H

Open Access

Symmetric stochastic inverse eigenvalue

problem

Quanbing Zhang

1*

, Changqing Xu

2

and Shangjun Yang

3

*Correspondence:

qbzhang@ahu.edu.cn

1Key Laboratory of Intelligent

Computing & Signal Processing, Ministry of Education, Anhui University, Hefei, Anhui, China Full list of author information is available at the end of the article

Abstract

Firstly we study necessary and sufficient conditions for the constant row sums symmetric inverse eigenvalue problem to have a solution and sufficient conditions for the symmetric stochastic inverse eigenvalue problem to have a solution. Then we introduce the concept of general solutions for the symmetric stochastic inverse eigenvalue problem and the concept of totally general solutions for the 3×3 symmetric stochastic inverse eigenvalue problem. Finally we study the necessary and sufficient conditions for the symmetric stochastic inverse eigenvalue problems of order 3 to have general solutions, and the necessary and sufficient conditions for the symmetric stochastic inverse eigenvalue problems of order 3 to have a totally general solution.

MSC: Primary 15C51; 15A18

Keywords: inverse eigenvalue problem; symmetric stochastic inverse eigenvalue problem; typical orthogonal matrix

1 Introduction

For a square matrixA, letσ(A) denote the spectrum ofA. Given ann-tuple= (λ, . . . ,λn)

of numbers, the problem of deciding the existence of ann×nmatrixA(of a certain class) withσ(A) =is called the inverse eigenvalue problem (of a certain class) which has for a long time been one of the problems of main interest in the theory of matrices.

Sufficient conditions for the existence of an entrywise nonnegative matrixAwithσ(A) = have been investigated by several authors such as Borobia [], Fiedler [], Kellogg [],

Marijuanet al.[] and Salzmann []. Soto and Rojo [] investigated the existence of cer-tain entrywise nonnegative matrices with a prescribed spectrum under cercer-tain conditions. Since stochastic matrices are important nonnegative matrices, it is surely important to in-vestigate the existence of stochastic matrices with a prescribed spectrum under certain conditions. It is well known [] that the nonnegative inverse eigenvalue problem and the stochastic inverse eigenvalue problem are equivalent problems. Hwanget al.[, ] and [] gave some interesting results for the symmetric stochastic inverse eigenvalue problem. In this work we mainly study the symmetric stochastic inverse eigenvalue problem.

For simplicity, we useSIEPfor Symmetric Inverse Eigenvalue Problem,SSIEPfor the Symmetric Stochastic Inverse Eigenvalue Problem; and we useCRS for Constant Row Sums, andSCRSIEPfor the SIEP with Constant Row Sums.

(2)

Since the spectrum of ann×nreal symmetric matrix is a set ofnreal numbers, we useto denote a realn-tuple (λ, . . . ,λn), whose entries are not necessarily distinct, for

the given possible spectrum of then×nSSIEP. For convenience, we may arrangein non-increasing order,i.e.,λ≥λ≥ · · · ≥λn. An-tupleisSS (resp. SCRS) realizableif

there exists ann×nsymmetric stochastic matrix (symmetric matrix with CRS)Awith σ(A) =. In this case,Ais called aSS (resp. SCRS) realizationof. For any two matrices AandBof ordernwe say thatBis permutationally similar toAifB=PAPT for some

permutation matrixPof ordern.

Throughout the paper, we denote byeRnthe all-ones column vector of dimensionn, we useunto denote the unit vector√ne, andInthe identity matrix of ordern.

The paper is organized as follows. In Section , we give necessary and sufficient condi-tions for the SCRSIEP. In Section , we give sufficient condicondi-tions for the SSIEP. In Section , we introduce the concept of general solution for SSIEP and the totally general solution for the × SSIEP and we give the necessary and sufficient conditions for the × and × SSIEP to have a general solution and the necessary and sufficient conditions for the × SSIEP to have totally general solution. Three examples are presented there.

2 Necessary and sufficient conditions for the SCRSIEP

The following result shows that any given set of real numbers is SCRS realizable.

Theorem . Let= (λ, . . . ,λn)be a real n-tuple.

() Ann×nsymmetric matrixAis a SCRS realization ofwith CRSλkif and only ifA

can be written as

A=Sdiag(λ, . . . ,λn)ST=λξξT+λξξT+· · ·+λnξnξnT, (.)

whereS= (ξ, . . . ,ξn)∈Rn×nis an orthogonal matrix(in column block form)with

ξk=un.

() The SCRSIEP foralways has a solutionAgiven by(.).

Proof () For the necessity, letAbe a SCRS realization ofwith CRSλk, thenσ(A) =

and Ae=λke, and that means un is the unit eigenvector of A associated with

eigen-valueλk. The symmetry ofAimplies thatAhas orthonormal eigenvectorsξ, . . . ,ξnsuch

thatAξj=λjξj,j= , , . . . ,n. LetS= (ξ, . . . ,ξn),D=diag(λ, . . . ,λn), thenSis an

orthog-onal matrix withξk=unsatisfyingAS=SDand henceA=SDSTis expressed as (.).

For the sufficiency, letAbe then×nsymmetric matrix given in (.). ThenAS=SD. It follows thatξj= ,Aξj=λjξjfor eachj= , , . . . ,nandAe=Anun=√nAξk=√nλkξk= √

nλkun=λke. Therefore,Ais a SCRS realization ofwith CRSλk.

() It suffices to prove that there always exists ann×northogonal matrixS= (ξ, . . . ,ξn)

withξk=unfor any givenk∈ {, , . . . ,n}since the matrixAgiven by (.) is a SCRS

realiza-tion ofwith CRSλkby (). Assume, without loss of generality, thatξ=un. Expanding

this vector ξinto an orthonormal base ofRn, we obtain the desired orthogonal matrix

S= (ξ, . . . ,ξn).

It is obvious that if a SCRS realizationAof a realn-tuple= (λ, . . . ,λn) has CRSλ,

(3)

there are a lot of (actually infinite forn> ) SCRS realizations of, each corresponds to a construction ofS.

Definition . Ann×ntypical orthogonal matrix is an orthogonal matrix with one col-umn equal to±un.

Remark . Ann×nmatrixSis typical orthogonal if and only ifPSQDis typical orthog-onal for any permutation matricesPandQand any unipotent diagonal matrixD i.e.such thatD=In.

Definition . Twon×ntypical orthogonal matricesMandH are equivalent ifH= PMQDfor some permutation matricesPandQand some diagonal unipotent matrixD.

Theorem . Let = (λ, . . . ,λn) (n> )be a given real n-tuple, S= (ξ, . . . ,ξn), S=

(ξ, . . . ,ξn)be typical orthogonal matrices,and A=Sdiag(λ, . . . ,λn)ST,A=Sdiag(λ, . . . , λn)S.We have:

() IfSandSare equivalent,thenAis permutationally similar toA.

() IfAis permutationally similar toAandλi=λjfor anyi=j,thenSandSare

equivalent.

Proof () IfS,Sare equivalent, thenS=PSQDfor some permutation matricesP,Qand diagonal unipotent matrixD. Denotingdiag(λ, . . . ,λn) byDwe have

A=SDS=PSQDDDQTSTPT=PSDSTPT=PAPT

and henceAis permutationally similar toA.

() IfAis permutationally similar toA, then there exists a permutation matrixPsuch thatSDS=A=PAPT=PSDSTPTfrom which it follows that

DSTPTS=STPTSD.

LetSTPTS=D= (d

ij), then the foregoing expression produces

λidij=dijλj, i,j= , . . . ,n. (.)

Now ifλi=λjfor anyi=j, then (.) yieldsdij=  for anyi=j. Therefore,STPTS=D

is a diagonal orthogonal matrix with each diagonal entry equal to±,i.e. D=I

n, which

meansDis a diagonal unipotent matrix. Finally, we haveS=PSD=PSInDand conclude

thatSandSare equivalent.

Remark . The symmetric matrixA=Sdiag(λ, . . . ,λn)ST is called a solution of

SCR-SIEP for= (λ, . . . ,λn) associated with the typical orthogonal matrixS. Then we can

say, by Theorem ., that two solutions of the SCRSIEP forassociated with equivalent typical orthogonal matrices are permutationally similar to each other.

(4)

Proof We prove it by contradiction and assume thatShas the block form: S= BD C ,

where  is ap×qzero matrix withp+q=n. Then theqcolumns ofCand the column of Dcorresponding to the vectorunform a system ofq+  linearly independent vectors of

dimensionq, which is impossible.

The next two lemmas present two nonequivalentn×ntypical orthogonal matricesSn,

Snfor anyn> :

Lemma . Letβ= (√n– )/(n– ),n> .Then the symmetric matrix

Sn= (sij) =

 √ n ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝    · · · 

 ( –n)β–  β · · · β

β ( –n)β–  · · · β

..

. ... ... . .. ...

β β · · · ( –n)β– 

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (.)

is a typical orthogonal matrix.

Proof Noticing thatβis the positive root of (n– )β+ β–  we can verify

ξjTξi=

, i=j,

, i=j.

A simple verification provides the next result.

Lemma . Let

Sn=sij=√ n ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝  –

n(n–)

n   · · · 

n

n(n–) –

n(n–)

n– . ..  

n(nn–) (n–)(nn–) . .. . .. ... ..

. ... ... . .. –

n

 

n(nn–) (n–)(nn–) · · · n·n

n(nn–) (n–)(nn–) · · ·

n · n  ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , (.) i.e.,

sij=

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩  √

n, j= ;

((n–j+ )(n–j+ ))–/, ij> ; –( ni

ni+)

/, j=i+ ; , otherwise.

(5)

Theorem ., together with Lemma . and Lemma ., produces the following result for the SCRSIEP.

Theorem . Let= (λ,λ, . . . ,λn) (n> ,λ≥λ≥ · · · ≥λn),D=diag(λ, . . . ,λn),and

Sn,Snare the typical orthogonal matrices given by(.), (.),respectively.Then both A=

SDSTand A=SDSTare symmetric matrices with CRS equal toλ

andσ(A) =.

There is only one × typical orthogonal matrix with the first column equal toun,i.e.

S=

 

 

 – 

.

But for anyn≥ there are infinite manyn×ntypical orthogonal matrices with the first column equal toun, as shown in the next lemma.

Lemma . Any×typical orthogonal matrix which having no entry equal to zero must be equivalent to

S(x) =

⎛ ⎜ ⎜ ⎜ ⎝

(x+x+)

x+

(x+x+)

 –

x+

(x+x+)

–x

(x+x+)

x

(x+x+)

x+

(x+x+)

⎞ ⎟ ⎟ ⎟

⎠ (.)

for some x∈(, ).

Proof LetSbe a × typical orthogonal matrix which has no zero entry. There is no loss of generality to assume thatSis equivalent to

S(u,v,x,y) =

⎛ ⎜ ⎜ ⎝

u v

 –(x+ )u yv

xu –(y+ )v

⎞ ⎟ ⎟ ⎠,

whereu,v,x,yare positive parameters. Normal orthogonality ofS(u,v,x,y) implies

u= 

(x+x+ ), v= 

(y+y+ ),

and

x( +y) +y( +x) = . (.)

It follows from (.) thaty= ( –x)/(x+ ) (> ), which yieldsx<  and

y+y+  = x+x+ /(x+ ).

(6)

Theorem . Any×typical orthogonal matrix must be equivalent to the matrix S(x) given by(.)for some x∈[, ].Any solution of the×SCRSIEP for= (λ,λ,λ)must be permutationally similar to

A(x) =S(x)diag(λ,λ,λ)ST(x) (.)

for some x∈[, ].

Proof LetSbe a × typical orthogonal matrix. ThenScannot have two zero entries in the same row or column by Lemma .. IfShas two zero entries, then the inner product of the two columns containing the two zero entries is surely nonzero, a contradiction to the orthogonality ofS. Therefore,Shas at most one zero entry. IfS has no zero entry, then the first assertion holds by Lemma .. IfShas exact one zero entry, thenSmust be equivalent to bothS= (u,β,ζ) andS= (u,ζ,β) withβ= (/√, –/√, ). Now the orthogonality ofSproducesζ = (/√, /√, –√/), and henceS=S(),S=S(). Therefore, the first assertion holds.

The second assertion holds by Theorem . and the first assertion.

Remark . It is interesting to note that the × typical orthogonal matrixSgiven by (.) is equivalent to bothS() andS(), andSgiven by (.) is equivalent toS(

–  ).

The following are some important × irreducible typical orthogonal matrices:

S=  ⎛ ⎜ ⎜ ⎜ ⎝      –    –     – ⎞ ⎟ ⎟ ⎟

⎠, S∗=

  ⎛ ⎜ ⎜ ⎜ ⎜ ⎝  –√    √  –  √    √   √  √   √   √  – √  ⎞ ⎟ ⎟ ⎟ ⎟ ⎠,

S(x) =

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝   –√       √  √

(x+x+)

x+

(x+x+)

  √

–(x+)

(x+x+)

–x

(x+x+)    √ x

(x+x+)

x–

(x+x+)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

, ≤x≤, (.)

S=  ⎛ ⎜ ⎜ ⎜ ⎝   √    –√   –  √  –  –√ ⎞ ⎟ ⎟ ⎟ ⎠, (.) S(x) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝  

x+

x+x+x+x+

x+x+

x+

x+x+

–x–x–

x+x+x+x+

x

x+x+

–x

x+x+

–x

x+x+x+x+

–(x+)

x+x+

–x

x+x+

x+x

x+x+x+x+

x

x+x+

x–

x+x+

⎞ ⎟ ⎟ ⎟ ⎟ ⎠. (.)

(7)

3 Sufficient conditions for the SSIEP

In this section we give two sets of sufficient conditions for the SSIEP. It is obvious thatA is a solution of the SSIEP if and only ifAis a solution of the SCRSIEP,ρ(A) =  andAis (entrywise) nonnegative. It is well known that

 =λ≥λ≥ · · · ≥λn≥– (.)

and

 +λ+· · ·+λn≥, (.)

are necessary for the non-increasingn-tupleto be SS realizable.

Theorem . Let= (λ= ,λ, . . . ,λn) (n> )satisfy(.)and

n+

λn(n– )+

λ

(n– )(n– )+· · ·+ λn

·≥. (.)

Thenis SS realizable.Moreover,the realizing matrix is(entrywise)positive if >λand (.)is a strict inequality.

Proof LetD=diag(,λ, . . . ,λn) andA=SnDSnT, whereSn= (sij) is then×ntypical

or-thogonal matrix given in (.). ThenAis symmetric with CRS equal to  andσ(A) =by Theorem ..

In order to prove the first assertion we only need to verify the nonnegativity ofAwhen (.) and (.) hold. For anyi= , , . . . ,n– , we have

aii=eTiSnDSnTei=si+λsi+· · ·+λi+si,i+

= n+

λ

n(n– )+· · ·+

λi

(n–i+ )(n–i+ )+

(n–i)λi+ (n–i+ );

ann=

n+

λn(n– )+

λ

(n– )(n– )+· · ·+ λn– · +

λn

·=an–,n–.

Since  =λ≥λ≥ · · · ≥λn, we obtain by simple calculation

ai–,i––aii=

ni

ni+ (λiλi+)≥

for anyi= , . . . ,n– . Therefore, condition (.) guarantees that

a≥ · · · ≥ann≥.

Now let us prove thataij≥ for all ≤i<jn. In fact, since

aij=eTi SnDSnTej=sisj+λsisj+· · ·+λnsinsjn

=  n+

λ

n(n– )+· · ·+

λi

(8)

we have (by (.))

aij

n+

λ

n(n– )+· · ·+

λi– (n–i+ )(n–i+ )

+ λi

(n–i+ )(n–i+ )– λi

ni+ 

=

n+

λ

n(n– )+· · ·+

λi– (n–i+ )(n–i+ )

λi ni+ 

n+

λ

n(n– )+· · ·+

λi– (n–i+ )(n–i+ )

λi– ni+ 

· · ·

≥ 

n–  n= 

for ≤i<jn. HenceAis nonnegative, and the first assertion is proved.

It is easy to see that if  >λand (.) is a strict inequality, thenaij>  for ≤ijn.

The second assertion is proved.

Remark . The sufficient condition (.) of SSIEP was first given by Theorem  of []. The first assertion of Theorem . is the same as the first assertion of Theorem  of [], the proof of the former presented here is simpler than the proof of the latter. Meanwhile, Fang [] disproved the second assertion made in Theorem  of [], which says thatis SS realizable by a positive matrixAif  >λ, together with (.) and (.). Our Theorem . shows that this assertion does hold if  >λand (.) is a strict inequality.

Theorem . Let= (λ= ,λ, . . . ,λn) (n> )satisfy(.), (.),andβ= √

n–

n– .If

 +β(λ+· · ·+λn–) +

(n– )β+ λn≥, (.)

 –(n– )β+ λ+β(λ+· · ·+λn)≥, (.)

 –(n– )β+ (λ+λ) +β(λ+· · ·+λn)≥, (.)

thenis SS realizable.Moreover,the realizing matrix is positive if(.), (.), (.),and (.)are all strict inequalities.

Proof LetD=diag(,λ, . . . ,λn) andA=SnDSnT, whereSn= (sij) is the typical orthogonal

matrix given in (.). ThenAis a symmetric matrix with CRS equal to  andσ(A) =by Theorem .. In order to prove the first assertion we only need to verify thatAis nonneg-ative under the conditions (.), (.), (.), and (.).

In fact (.) yieldsa=  +λ+· · ·+λn≥; (.) and (.) yield

aii=si+λsi+· · ·+λnsin

= n

 +(n– )β+ λi+β

k∈{/,i}

λk

≥

n

 +(n– )β+ λn+βn–

k= λk

(9)

forj= , . . . ,n; (.) and (.) yield

aj=ssj+· · ·+λnsnsjn

= n

 –(n– )β+ λj+β

k∈{/,j}

λk

≥ 

n

 –(n– )β+ λ+β

n

k= λk

≥

fori= , . . . ,n; and finally (.) and (.) yield

aij=sisj+· · ·+λnsinsjn

=  n

 –(n– )β+ (λi+λj) +β

k∈{/,i,j}

λk

≥ 

n

 –(n– )β+ (λ+λ) +β

n

k= λk

≥

for ≤in,j=i. SoAis nonnegative, and the first assertion is proved. The proof of the

second assertion is trivial.

4 The general solution and the totally general solution for the3×3SSIEP

Definition . Thegeneral solutionof the SSIEP for= (λ, . . . ,λn) is a set ofn×n

symmetric stochastic matrices such that each element of is a solution of the SSIEP for and each solution of the SSIEP foris permutationally similar to one element of .

Since any possible × typical orthogonal matrix is equivalent to

S=√ 

 –

 

,

any possible × symmetric stochastic matrix withσ(A) =={,λ}(|λ| ≤) must be permutationally similar to

Sdiag(,λ)ST= 

 +λ  –λ  –λ  +λ

(.)

by Theorem . and Theorem .. Therefore, the general solution of the × SSIEP for given= (,λ) exists if and only if|λ| ≤ and the solution is equivalent to the matrix given in (.).

Forn=  we have proved (Theorem .) that any solution of × SIEP with unit row sums must be permutationally similar to A(x) =S(x)diag(λ,λ,λ)ST(x) for somex∈ [, ], whereS(x) is given in (.). This motivates the following definition for the special case ofn= .

Definition . If A(x) = S(x)diag(,λ,λ)ST

(10)

Note that the general solution of the SSIEP for any= (,λ,λ) is a subset of{A(x), x≤}and they may not be equal. In other words, for the SSIEP, the totally general solution is the general solution, but the converse is not always true (see Example .).

Lemma . Let= (,λ,λ)satisfy(.)and(.);S(x) = (u,ξ,ξ) (≤x≤)be the ×typical orthogonal matrix given by(.),and A(x) =S(x)diag(,λ,λ)S(x)T.Then

A(x)is nonnegative if and only if + λ≥and – λ+λ≥.

Proof We have

A(x) =aij(x)

=uuT +λξξT+λξξT

= 

⎛ ⎜ ⎝

        

⎞ ⎟ ⎠+λ

α

⎛ ⎜ ⎝

 – –x x

– –x ( +x)–xxx –x–xx

⎞ ⎟ ⎠

+ λ α

⎛ ⎜ ⎝

( + x)  +x– x – – x– x  +x– x ( –x)– +x+x – – x– x – +x+x( +x)

⎞ ⎟

⎠, (.)

whereα= x+ x+  >  for allx[, ]. Then

αa(x) = ( + λ)x+ ( + λ)x+  + λ+λ,

αa(x) = ( + λ+λ)x+ ( + λ– λ)x+  + λ+λ,

αa(x) = ( + λ+λ)x+ ( + λ)x+  + λ,

αa(x) = ( – λ)x+ ( – λ+λ)x+  – λ+λ,

αa(x) = ( – λ)x+ ( + λ– λ)x+  – λ,

αa(x) = ( – λ+λ)x+ ( – λ+λ)x+  – λ.

Since α(a(x) –a(x)) = ( –x)λ

– ( –x)λ= ( –x)(λ–λ)≥, we conclude a(x)≥a(x), ≤x≤. Then if  + λ+λ>  we have

min

≤x≤αa(x)≥minxRαa(x)

=  + λ–

( + λ) ( + λ+λ)

=( + λ)( + λ)  + λ+λ

and if  + λ+λ≤, we have

min

≤x≤αa(x) =min

( + λ), ( +λ+ λ)

.

Note thatλ≥λleads to  + λ+λ≥ +λ+ λ≥( + λ) and  + λ≥ + λ. If  + λ≥, then

min

≤x≤αa(x)≥min

( + λ),

( + λ)( + λ)  + λ+λ

(11)

On the other hand, ifa(x)≥ for allx∈[, ], then

 + λ= 

αa()≥ 

min≤x≤αa(x)≥.

Therefore,a(x)≥⇔ + λ≥.

Since  – λ≥ and  + λ+λ≥( +λ+λ)≥ by (.) we have

min

≤x≤a(x) =min≤x≤  α

( + λ+λ)x+ ( + λ– λ)x+  + λ+λ

≥.

Now  + λ– λ≥ – λ≥ – λ+λprovidesa(x)≥a(x), ≤x≤. Moreover, α(a(x) –a(x)) = ( –x)(λ

–λ)≥ yields

mina(x),a(x),a(x)

=a(x), ≤x≤.

Now if  – λ> , then

min

≤x≤αa(x)≥minxRαa(x)

=  – λ+λ–

( – λ+λ) ( – λ)

=( +λ– λ)( – λ+λ) ( – λ)

.

If  – λ= , then

αa(x) = ( – λ+λ)( +x).

Since  +λ– λ≥,  +x> , we have

a(x)≥ ⇔  – λ+λ≥.

Therefore,A(x) is nonnegative if and only if  + λ≥ and  – λ+λ≥.

Since  + λ≥⇒ +λ+λ≥, Theorem . and Lemma . produce the following result.

Theorem . The totally general solution of the SSIEP for= (,λ,λ) (≥λ≥λ≥–) exists if and only if

 + λ≥ and  – λ+λ≥, (.)

and the totally general solution is given by(.).

As we know, a quadratic real functionf(x) =ax+bx+cwitha=  has two different real zeros if and only if its discriminant=b– ac> , and the two zeros arex

±=–b± √

(12)

>  anda>  (< ) thenf(x)≥ on [–∞,x–]∪[x+, +∞]([x–,x+]). The quadratic functionαa(x) = ( + λ+λ)x+ ( + λ)x+  + λhas two real different zeros:

u±=– – λ± √

–( + λ)( + λ)  + λ+λ

(.)

if and only if  + λ<  <  + λ; and αa(x) = ( – λ)x+ ( – λ+λ)x+  – λ+λ has two real different zeros:

v±=– + λ–λ± √

–( – λ+λ)( +λ– λ)

( –λ) (.)

if and only if  – λ+λ<  <  +λ– λ.

Theorem . Let= (,λ,λ)satisfy(.)and(.).Then the general solution of SSIEP forexists if and only if

 +λ+ λ≥, (.)

and the general solution is given by(.)with≤x≤when + λ≥and – λ+λ≥ ;with v+x≤when + λ≥and – λ+λ< ;with u+x≤when + λ<  and – λ+λ≥;withmax{u+,v+} ≤x≤when + λ< and – λ+λ< .

Proof Let A(x) = (aij(x)) =S(x)diag(,λ,λ)S(x)T, where S(x) is given by (.). Let

Nij(x)(≤i,jn)⊂[, ] denote the ‘nonnegative interval’ ofaij(x) such thataij(x)≥

if and only ifxNij(x). ThenA(x) for somex∈[, ] is a solution of the SSIEP forif and

only if =i,j=,...,nNij(x)=∅by Theorem . and Theorem .. SinceN(x)N(x),

N(x)N(x)N(x) andN(x) = [, ] by Lemma ., =N(x)N(x).

If (.) does not hold, then  + λ≤ +λ+ λ< . Moreover,  + λ+λ<  implies αa(x) <  for allx∈[, ], and  + λ+λ≥ >  +λ+ λimplies

αa(x)≤( + λ+λ)x+ ( + λ)x+  + λ

=  + λ+ λ+  + λ= ( +λ+ λ) < ,

for allx∈[, ]. Therefore,N(x) =∅, which yields =NN(x) =∅. This proves the necessity.

To prove the sufficiency, it suffices to show that (.) implies =∅. If  + λ≥, then N(x) = [, ] by the proof of Lemma .. If  + λ< , then  + λ< ≤( +λ+ λ) <  + λ, which producesu–<  <u+and henceN(x) = [, ]∩([u+, +∞)∪[–∞,u–]) = [u+, ]. Similarly if  – λ+λ≥, then N(x) = [, ] by the proof of Lemma .. If  – λ+λ< , thenλ>λ,  – λ+λ< ≤ – λ<  +λ– λ, which producesv–<  <v+and henceN(x) = [, ]∩([v+, +∞)∪[–∞,v–]) = [v+, ]. So =N(x)N(x) = [u+, ]∩[v+, ].

Now we prove, by contradiction, that condition (.) impliesmax{u+,v+} ≤, which pro-duces [u+, ]∩[v+, ] = [max{u+,v+}, ]=∅, and then the proof of sufficiency is completed. As a matter of fact, ifu+> , then  + λ+λ< – – λ+

(13)

Sou+>  is equivalent to

( +λ+λ) <

–( + λ)( + λ),

or

λ+ λλ+ λ+ λ+ λ+  < .

But this is impossible because the left side of the last inequality is equal to ( + λ+ λ)( +λ+ λ)≥( +λ+ λ)≥ by (.). Similarly ifv+> , then ( –λ–λ) <

–( – λ+λ)( +λ– λ) by (.). Sou+>  is equivalent to

(λλ–λ–λ+ ) < .

But this is impossible because the left side of the last inequality is equal to ( –λ)( –λ),

which cannot be negative.

Remark . Theorem  of [] shows that the real triple ={,λ,λ}with |λi| ≤, i= ,  is SS realizable if and only if

 + λ+λ≥ and  +λ+ λ≥. (.)

It is clear that condition (.) and|λi| ≤,i= ,  is equivalent to conditions (.), (.), and (.).

Example . Since= (, , –.) satisfies condition (.) of Theorem ., the totally general solution of the SSIEP foris

A(x) =  αM(x)

, ≤x≤,

where

M(x) =

⎛ ⎜ ⎝

α– .( + x) α– .( +x– x) α+ .( + x+ x)

α– .( –x)α+ .( –xx)

∗ ∗ α– .( +x)

⎞ ⎟ ⎠,

α= x+ x+  > , ≤x≤, and the entries marked by ‘∗’ are determined by the sym-metry property ofM(x).

Moreover, since satisfies condition (.), Theorem . provides a solution of this SSIEP:

A∗=S

⎛ ⎜ ⎝

  

  

  –.

⎞ ⎟ ⎠ST=

⎛ ⎜ ⎝

/ / / / / / / / 

(14)

Note that sincealso satisfies the condition of Theorem  of [], a symmetric stochastic matrixAwithσ(A) =can be found by the method given in [] to be

⎛ ⎜ ⎝

/ / /

/ /, ,/, / ,/, /,

⎞ ⎟ ⎠,

which is not equivalent toA∗. So our Theorem . and Theorem  of [] are really different.

Remark . If we find the general solution for the SSIEP, then we find all the possible solutions (only different by a permutationally similar transformation) and we are able to choose some interesting solution from them. As an example, for Example . it is inter-esting to know:among all the possible SS realizations of, which one has the greatest (smallest) entry? and what is the value of this greatest (smallest) entry?

Using the expression of the general solution:A(x) =αM(x), x≤, we can easily find that the (, ) entry ofA() equal to(the (, ) entry ofA() equal to > ) is the greatest (least) entry among all the entries of all the possible SS realizations of. Therefore, any possible SS realization ofis always a positive matrix whose entries belong to the interval [,].

Example . Since= (, , –.) does not satisfy the conditions of both Theorem . and Theorem ., the totally general solution of the SSIEP for  does not exist, and A(√– ) is not a solution. But does satisfy conditions (.) and (.), and this SSIEP has a solution

A∗=

⎛ ⎜ ⎝

/ / /

/ / / / / /

⎞ ⎟ ⎠

by Theorem ..

Now we use Theorem . to find the general solution of this SSIEP as follows. Since

 + λ< ≤ + λ+λand  <u+=

––λ+√(+λ)(––λ)

+λ+λ = . <  we haveS(x) = [u+, ]=∅; since  – λ+λ≥ we haveS(x) = [, ] and =S= [u+, ]. Finally the general solution is

⎧ ⎪ ⎨ ⎪ ⎩A(x) =

 α

⎛ ⎜ ⎝

α– .( + x) α– .( +x– x) α+ .( + x+ x)

α– .( –x)α+ .( –xx)

∗ ∗ α– .( +x)

⎞ ⎟ ⎠ ⎫ ⎪ ⎬ ⎪ ⎭

withx∈[., ]. It is interesting to note that the preceding solutionA∗is permuta-tionally similar toA().

(15)

permutation-ally similar to each other:

A=

⎛ ⎜ ⎜ ⎜ ⎝

. . . . . . . . . . . . . . . .

⎞ ⎟ ⎟ ⎟

⎠, A∗=

⎛ ⎜ ⎜ ⎜ ⎝

. . . . . . . . . . . . . . . .

⎞ ⎟ ⎟ ⎟ ⎠.

This shows that if the condition of Theorem .() is not satisfied,i.e.contains two equal elements, then for two typical orthogonal matricesS, S of ordern, the relation showing that S is equivalent toS cannot be guaranteed by the relation showing that Sdiag(λ, . . . ,λn)STis permutationally similar toSdiag(λ, . . . ,λn)S.

Using the typical orthogonal matrixSgiven in (.) we have the following corollary.

Corollary . If

 +λ+ λ≥, (.)

then

A=Sdiag(,λ,λ,λ)S

=  

⎛ ⎜ ⎜ ⎜ ⎝

 +λ+ λ  +λ– λ  –λ  –λ  +λ– λ  +λ+ λ  –λ  –λ

 –λ  –λ  +λ+ λ  +λ– λ  –λ  –λ  +λ– λ  +λ+ λ

⎞ ⎟ ⎟ ⎟ ⎠

is a solution of the×SSIEP for= (,λ,λ,λ) (≥λ≥λ≥λ≥–).

Corollary . If

 + λ≥,  +λ+ λ≥,  +λ– λ+ λ≥, (.)

then the×SSIEP for= (,λ,λ,λ) (≥λ≥λ≥λ≥–)has solutions with one parameter x:

A(x) =S(x)diag(,λ,λ,λ)ST(x), ≤x≤.

Proof LetA(x) =S(x)diag(,λ,λ,λ)ST

 (x), ≤x≤ withS(x) given by (.). Then for ≤x≤,A(x) is a × symmetric matrix with unit row sums realizingby Theorem . and

A(x) =

⎛ ⎜ ⎜ ⎜ ⎝

+λ 

–λ 

–λ 

–λ  –λ

 –λ

A(x)

–λ 

(16)

whereA(x) = (, , )T(, , ) +S

(x)diag(λ,λ,λ)ST(x) withS(x) =S(x)diag(, , )× ST

(x). LetS(x)diag(λ,λ,λ)ST(x) = (aij(x)), then a direct calculation produces

αa(x) =

λ  + λ

x+

λ  + λ

x+λ

 + λ+λ,

αa(x) =

λ

 + λ+λ

x+

λ  – λ

x+λ

 + λ+λ,

αa(x) =

λ

 + λ+λ

x+

λ  + λ

x+λ  + λ,

αa(x) =

λ  – λ

x+

λ

 – λ+λ

x+λ

 – λ+λ,

αa(x) =

λ  – λ

x+

λ

 + λ– λ

x+λ  – λ,

αa(x) =

λ

 – λ+λ

x+

λ

 – λ+λ

x+λ  – λ.

By the analysis used in the proof of Lemma . we conclude that forx∈[, ]

Minx[,]S(x)diag(λ,λ,λ)ST(x)

=Mini<jnaij(x)≥Min

λ  + λ,

λ

 – λ+ λ

. (.)

Now condition (.) implies Min{λ  + λ,

λ

 – λ + λ} ≥–, and hence A(x) = (, , )T(, , ) +S(x)diag(λ,λ,λ)ST(x) is a nonnegative matrix. Finally, since  –λ≥ and  + λ≥ by (.) we see thatA(x) is also a nonnegative matrix, and hence a

sym-metric stochastic matrix withσ(A) =.

Open problems What are the necessary and sufficient conditions of the × SSIEP to have a solution? What is the general solution of this inverse eigenvalue problem when it has a solution?

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significently in writing this paper. All authors read and approved the final manuscript.

Author details

1Key Laboratory of Intelligent Computing & Signal Processing, Ministry of Education, Anhui University, Hefei, Anhui, China. 2Department of Applied Mathematics, Suzhou University of Science and Technology, Suzhou, Jiangsu, China.3School of

Mathematical Sciences, Anhui University, Hefei, Anhui, China.

Acknowledgements

Q Zhang is supported by the National Natural Science Foundation of China (Nos. 61301296, 61377006, 61201396) and the National Natural Science Foundation of China - Guangdong Joint Fund (No. U1201255). C Xu is supported by the China State Natural Science Foundation Monumental Project (No. 6119010) and the China National Natural Science Foundation Key Project (No. 61190114). The financial support of them is gratefully acknowledged.

Received: 9 October 2014 Accepted: 18 May 2015 References

(17)

3. Kellogg, RB: Matrices similar to a positive of essentially positive matrix. Linear Algebra Appl.4, 191-204 (1971) 4. Marijuan, C, Pisonero, M, Soto, RL: A map of sufficient conditions for the real nonnegative inverse eigenvalue

problem. Linear Algebra Appl.426, 690-705 (2007)

5. Salzmann, FL: A note on eigenvalues of nonnegative matrices. Linear Algebra Appl.5, 329-338 (1972)

6. Soto, R, Rojo, O: Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem. Linear Algebra Appl.416, 844-856 (2006)

7. Johnson, CR: Row stochastic matrices similar to doubly stochastic matrices. Linear Multilinear Algebra10, 113-130 (1981)

8. Hwang, SG, Pyo, SS: The inverse eigenvalue problem for symmetric doubly stochastic matrices. Linear Algebra Appl.

379, 77-83 (2004)

9. Lei, Y, Xu, W, Lu, Y, Niu, Y, Gu, X: On the symmetric doubly stochastic inverse eigenvalue problem. Linear Algebra Appl.

445, 181-205 (2014)

10. Zhang, Q, Xu, C, Yang, S: On the inverse eigenvalue problem for irreducible doubly stochastic matrices of small orders. Abstr. Appl. Anal.2014, Article ID 902383 (2014). doi:10.1155/2014/902383

11. Perfect, H, Mirsky, L: Spectral properties of doubly stochastic matrices. Monatshefte Math.69, 35-57 (1965) 12. Fang, M: A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices. Linear Algebra Appl.

Figure

Updating...

References