The inverse eigenvalue problem is a classical and difficult problem in matrix theory. In the case of real spectrum, we first present some sufficient conditions of a real r-tuple (for r=2; 3; 4; 5) to be realized by a symmetric stochastic matrix. Part of these conditions is also extended to the complex case in the case of complex spectrum where the realization matrix may not necessarily be symmetry. The main approach throughout the paper in our discussion is the specific construction of realization matrices and the recursion when the targeted r-tuple is updated to a (r+1)-tuple.
1. Introduction
For a square matrix A, let σ(A) denote the spectrum of A. Given an n-tuple Λ=(λ1,…,λn) of numbers, real or complex, the problem of deciding the existence of a nonnegative matrix A with σ(A)=Λ is called the nonnegative inverse eigenvalue problem (NIEP) 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 positive matrix A with σ(A)=Λ have been investigated by many authors [1–14]. The case n=2 is trivial. The problem has been solved for n=3 by Loewy and London [6]. The cases n=4 and n=5 have been solved for matrices with trace zero by Reams [10] and Laffey and Meehan [5], respectively. So, for real spectra, complete constructive solutions to NIEP are available for n≤4. For the case of nonreal spectra for n=4, complete solutions are available through the work of Laffey and Meehan [5], independently, and that of Torre-Mayo et al. [12] by analyzing coefficients of the characteristic polynomial. EBL digraphs, and for n=2,3,4,5, complete solutions are available through the work of Nazari and Sherafat [8].
An n×n nonnegative matrix A=(aij) is called a row stochastic matrix if ∑j=1naij=1, i=1,…,n; A is called a doubly stochastic matrix if ∑j=1naij=1, i=1,…,n, and ∑i=1naij=1, j=1,…,n. Since row stochastic and doubly stochastic matrices are important nonnegative matrices, it is surely important to investigate the existence of row or doubly stochastic matrices with prescribed spectrum under certain conditions. We call this special NIEP the row or doubly stochastic inverse eigenvalue problem (RSIEP or DSIEP). Hwang and Pyo [3] gave some interesting results for the symmetric DSIEP.
An n-tuple Λ=(λ1,…,λn) is nonnegative (doubly stochastic) realizable if there exists an n×n nonnegative (doubly stochastic) matrix A with σ(A)=Λ. In this case, we say A is a nonnegative (doubly stochastic) realization of Λ or the nonnegative (doubly stochastic) matrix A that realizes the n-tuple Λ.
A nonincreasing n-tuple Λ=(λ1,…,λn) is called S-feasible if it satisfies
∑j=1nλj≥0;
1≡λ1≥|λj| for all j=2,…,n.
Throughout the paper, we denote the spectrum of A by σ(A); the spectrum radius of A by ρ(A); the all-ones column vector of n dimensions by e∈Rn. We use un for the n dimensional normalized vector (1/n)eT and In for the identity matrix of order n.
Theorem NN (see [15]). Let A be an irreducible nonnegative matrix of order n. Then, we have
ρ(A)>0 and ρ(A)∈σ(A);
A(x)=ρ(A)x for somex inRn, and the null space of A-ρ(A)In is of dimension 1;
|λ|<ρ(A) for any λ≠ρ(A),λ∈σ(A).
In this paper, we study DSIEP of order n∈{2,3,4,5}. In Section 2, we present some sufficient conditions for the DSIEP for a given real n-tuple. In Section 3, we present some sufficient conditions for the DSIEP for a given nonreal complex n-tuple where the realization matrix may not necessarily be symmetric.
2. The Case of Real SpectrumLemma 1.
If 1>λ2≥-1, then there is a 2×2 irreducible doubly stochastic matrix realizing Λ={1,λ2}.
Proof.
It is easy to verify that the following 2×2 irreducible doubly stochastic matrix:
(1)12(1+λ21-λ21-λ21+λ2)
realizes Λ={1,λ2}.
In this section, we present a theorem that is analogy to Theorem 2.1 of [8]. The theorem is used to construct an n×n irreducible symmetric stochastic realization of a given n-tuple with designed conditions.
Theorem 2.
For any integer m≥2 if B is an m×m irreducible doubly stochastic matrix with σ(B)={1,μ2,…,μm} and 1>λ≥-1, then there exists an (m+1)×(m+1) irreducible doubly stochastic matrix C such that σ(C)={1,cμ2,…,cμm,μm+1}, where c=1-(1-λ)/2m=(2m-1+λ)/2m>0; μm+1=((m+1)λ+m-1)/2m.
Proof.
We know, by Theorem NN, that ρ(B)=1 is a simple eigenvalue of B and um is the unique normalized positive eigenvector associated with 1 such that umTum=1, Bum=um, umTB=umT. Now we can find an m×(m-1) matrix V1 such that Y1=(um,V1) is a unitary matrix. Then,
(2)B1=Y1*BY1=(umTumumTBV1V1*umV1*BV1)=(1*0B^),
where (V1*um)*=umTV1=0, B^=V1*BV1 and it is not necessary to know the value of each entry in the location remarked by “*.” Since σ(B1)=σ(B)={1,μ2,…,μm}, we have σ(B^)={μ2,…,μm}. By Schur’s unitary triangularization theorem (see [15]), there exists a unitary matrix V2 of order m-1, such that V2*B^V2=T^B, where T^B is a upper triangular matrix and σ(B^) is the set of all diagonal entries of T^B. Now for the m×m unitary matrix Y2=(1)⊕V2, we have
(3)Y2*B1Y2=Y2*(Y1*BY1)Y2=(Y1Y2)*B(Y1Y2)=Y*BY,
where Y=Y1Y2=(um,V1V2)=(um,T) is unitary with T=V1V2 and
(4)YY*=umumT+TT*=Im,Y*Y=(umTumumTTT*umT*T)=(1)⊕Im-1.
Therefore,
(5)umTT=0,T*um=0,T*T=Im-1,Y*BY=(umTBumumTBTT*BumT*BT)=(1*0T*BT),
where T*Bum=T*um=0 by (5) and
(6)σ(T*BT)=σ(Y*BY)-{1}=σ(B)-{1}={μ2,…,μm}.
It is easy to verify that the following (m+1)×(m+1) matrix:
(7)C=(1+λ21-λ2mumT1-λ2mum(1-(1-λ)/2m)B)=(1+λ21-λ2mumT1-λ2mumcB)
is a doubly stochastic matrix. Let β=(1/2,-1/2)T; then the following (m+1)×(m+1) matrix:
(8)Z=(u2T0umβTT)
is a unitary matrix, since
(9)ZZ*=(u2Tu200TT*)=(100Im)
by u2Tβ=0, βTu2=0, and umumT+TT*=Im. In addition using umTum=1 and T*Bum=T*um=0, we have
(10)Z*CZ=(1+λ2u2u2T+1-λ2mβu2T+1-λ2mu2βT+cββT00cT*BT)=(C100cT*BT),
where
(11)C1=1+λ2u2u2T+1-λ2mβu2T+1-λ2mu2βT+cββT=(1+λ41+λ41+λ41+λ4)+(1-λ8m1-λ8m-1-λ8m-1-λ8m)+(1-λ8m-1-λ8m1-λ8m-1-λ8m)+(c2-c2-c2c2).
Therefore, σ(C)=σ(C1)∪σ(cT*BT)=σ(C1)∪{cμ2,…,cμm}. Since 0<c<1, we have |cμi|<1, i=2,…,m and hence 1∉{cμ2,…,cμm}. In addition, 1∈σ(C) (for C is a doubly stochastic matrix) implies that 1∈σ(C1). So the spectrum of C1 is σ(C1)={1,tr(C1)-1}={1,μm+1}. Now a direct calculation produces the following:
(12)μm+1=tr(C1)-1=(m+1)λ+3m-12m-1=(m+1)λ+m-12m.
Finally, the irreducible doubly stochastic matrix C has the desired spectrum σ(C)={1,μ2,…,μm+1}.
The following result is obtained by a similar argument used in Theorem 2.
Corollary 3.
Let B be an irreducible row (symmetric) stochastic matrix with m-tuple Λ={1,μ2,…,μm} as its spectrum. Denote Λ¯={1,cμ2,…,cμm,μm+1=((m+1)λ+m-1)/2m}, where c=(2m-1+λ)/2m. Then Λ¯ can be realized by an irreducible row (symmetric) stochastic matrix.
Notice that a real n-tuple Λ={λ1,λ2,…,λn} is realized by an irreducible doubly stochastic matrix A only if max{λ1,…,λn}=1=ρ(A) is a simple eigenvalue of A. For convenience, we always assume that 1=λ1>λ2≥λ3≥⋯≥λn≥-1.
Corollary 4.
If the real triple Λ={1,λ2,λ3}(1>λ2≥λ3≥-1) satisfies
(13)2+λ2+3λ3≥0,
then Λ is realized by a symmetric irreducible stochastic matrix.
Proof.
Assume that Λ satisfies Condition (13) which implies that λ2≥-0.5. Let m=2, λ2=μm+1=((m+1)λ+m-1)/2m=(3λ+1)/4; then λ=(4λ2-1)/3∈[-1,1), c=(2m-1+λ)/2m=(3+λ)/4=(11+4λ2)/12>0. Let cμ2=λ3 and
(14)B=12(1+λ31-λ31-λ31+λ3).
Then B is an irreducible doubly stochastic matrices with σ(B)={1,λ3} by Lemma 1 and
(15)cB=c(1+λ3/c21-λ3/c21-λ3/c21+λ3/c2)=(c+λ32c-λ32c-λ32c+λ32)=(8+4λ2+12λ3248+4λ2-12λ3248+4λ2-12λ3248+4λ2+12λ324)=(2+λ2+3λ362+λ2-3λ362+λ2-3λ362+λ2+3λ36)
is nonnegative. Finally, the matrix of order m+1=3 in (7)
(16)C=(1+λ21-λ22u2T1-λ22u2cB)=(1+2λ231-λ231-λ231-λ232+λ2+3λ362+λ2-3λ361-λ232+λ2-3λ362+λ2+3λ36)
is irreducible symmetric stochastic with σ(C)={1,cμ2,μ3}=Λ by Theorem 2.
Remark 5.
Theorem 14 of [8] shows that Condition (13) is sufficient and necessary for a real triple Λ={1,λ2,λ3} to be doubly stochastic realizable.
Corollary 6.
If a real feasible nonincreasing 4-tuple Λ={1,λ2,λ3,λ4} satisfies
(17)3+λ2+2λ3+6λ4≥0,
then Λ is realized by a symmetric irreducible stochastic matrix.
Proof.
Assume that Λ satisfies Condition (17). Then 0≤3+λ2+2λ3+6λ4≤3+9λ2 yields λ2≥-1/3. Let m=3, λ2=μm+1=((m+1)λ+m-1)/2m=(4λ+2)/6; then λ=(3λ2-1)/2∈[-1,1), c=(2m-1+λ)/2m=(5+λ)/6=(3+λ2)/4>0. It is clear that λ4/c=4λ4/(3+λ2)≥-1 since 4λ4/(3+λ2)<-1 would produce 3+λ2+4λ4<0, which, together with (17), yields
(18)3+λ2+2λ3+6λ4>3+2λ2+4λ4⟹λ3+λ4>0.
It follows that 3+2λ2+4λ4≥3(1+λ4)+(λ3+λ4)>0, which is a contradiction. So 1>4λ2/(3+λ2)≥4λ3/(3+λ2)=λ3/c≥λ4/c≥-1. Since 2+λ3/c+3λ4/c=(1/2c)(4c+2λ3+6λ4)=(1/2c)(3+λ2+2λ3+6λ4)≥0 by (17), the following matrix:
(19)B=(1+2λ3/c31-λ3/c31-λ3/c31-λ3/c32+λ3/c+3λ4/c62+λ3/c-3λ4/c61-λ3/c32+λ3/c-3λ4/c62+λ3/c+3λ4/c6)
is irreducible doubly stochastic by whom {1,λ3/c,λ4/c}={1,μ2,μ3} is realized by Corollary 4 and
(20)cB=(3+λ2+8λ3123+λ2-4λ3123+λ2-4λ3123+λ2-4λ3123+λ2+2λ3+6λ463+λ2+2λ3-6λ463+λ2-4λ3123+λ2+2λ3-6λ463+λ2+2λ3+6λ46)
is nonnegative. Finally, the matrix of order m+1=4 in (7)(21)C=(1+λ21-λ23u3T1-λ23u3cB)=(1+3λ241-λ241-λ241-λ241-λ243+λ2+8λ3123+λ2-4λ3123+λ2-4λ3121-λ243+λ2-4λ3123+λ2+2λ3+6λ4123+λ2+2λ3-6λ4121-λ243+λ2-4λ3123+λ2+2λ3-6λ4123+λ2+2λ3+6λ412)
is irreducible symmetric stochastic with σ(C)={1,cμ2,cμ3,μ4}=Λ by Theorem 2.
Using this recursive method, we can prove the following result.
Corollary 7.
Let Λ={1,λ2,λ3,λ4} be S-feasible and satisfies
(22)12+3λ2+5λ3+10λ4+30λ5≥0,
then Λ can be realized by a symmetric irreducible stochastic matrix.
Proof.
Assume that Λ satisfies Condition (22) which implies that λ2≥-1/4. Let m=4,λ2=μm+1=((m+1)λ+m-1)/2m=(5λ+3)/8; then λ=(8λ2-3)/5∈[-1,1),c=(2m-1+λ)/2m=(7+λ)/8=(4+λ2)/5>0. Now under Condition (22) using the same recursive method and Theorem 2, we can construct the following irreducible symmetric stochastic matrix:(23)(1+4λ251-λ251-λ251-λ251-λ251-λ254+λ2+15λ3204+λ2-5λ3204+λ2-5λ3204+λ2-5λ3201-λ254+λ2-5λ32012+3λ2+5λ3+40λ46012+3λ2+5λ3-20λ46012+3λ2+5λ3-20λ4601-λ254+λ2-5λ32012+3λ2+5λ3-20λ46012+3λ2+5λ3+10λ4+30λ56012+3λ2+5λ3+10λ4-30λ5601-λ254+λ2-5λ32012+3λ2+5λ3-20λ46012+3λ2+5λ3+10λ4-30λ56012+3λ2+5λ3+10λ4+30λ560),
by whom Λ is realized.
The next result shows that under some stronger conditions, we can construct a 3×3 nonsymmetric irreducible doubly stochastic matrix to realize the given real triple.
Proposition 8.
Let Λ={1,λ2,λ3} satisfy 1>λ2≥λ3≥-1. If
(24)λ2λ3+2λ2+2λ3+1>0,
then Λ is realized by a 3×3 nonsymmetric irreducible doubly stochastic matrix.
Proof.
If Λ satisfies Condition (24), then construct the following bunch of 3×3 irreducible matrices with one parameter p and trace of 1+λ2+λ3 whose row sums are all equal to 1:
(25)A(p)=(p1-p01+λ2+λ3-3pp2p-λ2-λ32p-λ2-λ301+λ2+λ3-2p).
Then det(A(p))=-6p2+3(2+λ2+λ3)p-1-2(λ2+λ3) and det(A(p))=λ2λ3 become p2-((2+λ2+λ3)/2)p+(λ2λ3+2(λ2+λ3)+1)/6=0 which has a positive zero as follows:
(26)p*=2+λ2+λ34-(2+λ2+λ34)2-λ2λ3+2λ2+2λ3+16=2+λ2+λ34-2(1-λ2)2+2(1-λ3)2+(λ2-λ3)248.
Moreover,
(27)p*<2+λ2+λ32<1,(28)2p*-λ2-λ3=1-λ22+1-λ32-2(1-λ2)2+2(1-λ3)2+(λ2-λ3)248≥(1-λ2)2+(1-λ3)24-2(1-λ2)2+2(1-λ3)2+(λ2-λ3)248=11(1-λ2)2+11(1-λ3)2+(λ2-λ3)248-2(1-λ2)2+2(1-λ3)2+(λ2-λ3)248≥0.
In addition, since
(29)6(1-λ2)2+6(1-λ3)2+3(λ2-λ3)2-(λ2+λ3-2)2=4(1-λ3)2+4(λ2-λ3)2+4(λ2-λ3)2>0,
we have
(30)1+λ2+λ3-3p*=λ2+λ3-24+146(1-λ2)2+6(1-λ3)2+3(λ2-λ3)2>0.
Therefore, A(p*) is an irreducible doubly stochastic matrix with tr(A(p*))=1+λ2+λ3,det(A(p*))=λ2λ3 and hence σ(A(p*))=Λ.
Remark 9.
Corollary 6 produces the sufficient condition (13) for an irreducible symmetric stochastic matrix of order 3 to have the prescribed real spectrum, and Proposition 8 produces the sufficient condition (24) for an irreducible nonsymmetric doubly stochastic matrix of order 3 to have the prescribed real spectrum. Note that Condition (24) implies Condition (13) because 2+λ2+3λ3=1+2λ2+2λ3+λ2λ3+(1-λ2)(1+λ3)≥0 if 1+2λ2+2λ3+λ2λ3≥0.
Theorem M (see [7]).
Let Λ′={1,λ2′,λ3′,λ4′} with -1≤λ2′,λ3′,λ4′≤1. Then Λ′ is realized by a symmetric doubly stochastic matrix with zero trace if and only if 1+λ2′+λ3′+λ4′=0, and when Λ′ satisfies the condition, the matrix is
(31)B=(01+λ2′1+λ3′1+λ4′1+λ2′01+λ4′1+λ3′1+λ3′1+λ4′01+λ2′1+λ4′1+λ3′1+λ2′0).
Corollary 10.
Let Λ={1,λ2,λ3,λ4,-1/4},-1≤λ2,λ3,λ4≤1 and c=3/4. If
(32)-1≤λ2c,λ3c,λ4c≤1,λ2+λ3+λ4=-c,
then Λ is realized by a symmetric doubly stochastic matrix with zero trace.
Proof.
Let λk′=λk/c,k=2,3,4. If (32) holds, then we have -1≤λ2′,λ3′,λ4′≤1 and 1+λ2′+λ3′+λ4′=0. So {1,λ2′,λ3′,λ4′}={1,λ2/c,λ3/c,λ4/c} is realized by the 4×4 doubly stochastic matrix B given in (31) by Theorem M. Let m=4,-1/4=λ5=(m+1+λ)/2m=(5+λ)/8; then λ=-1, c=(2m-1+λ)/2m=3/4. Now {1,cλ2,cλ3,cλ4}=Λ is realized by the 5×5 matrix (given in (7)) as follows:
(33)C=(1+λ21-λ4u4T1-λ4u4cB)=(012u4T12u4cB),
that is an irreducible doubly stochastic with zero by Theorem 2.
3. The Case of Complex Spectrum
Given a circulant doubly stochastic matrix, it is easy to obtain its spectrum (see Lemma 11). In this section, we use this result to construct an IDS (irreducible doubly stochastic) matrix to realize a given complex triple containing a pair of conjugate complex numbers with some additional conditions. This matrix is used together with Theorem 2 to construct an IDS matrix to realize a given complex 4-tuple and a 5-tuple containing exactly a pair of conjugate complex numbers with special conditions in a recursive method. Also constructed is an IDS realization of a given complex 5-tuple, which contains two pairs of conjugate complex numbers with special conditions.
The following result is well known. We give a short proof for completeness.
Lemma 11.
In the complex plane, let Ωn be the regular polygon whose vertices are all the nth roots of unity as follows: qk=
cos
(2kπ/n)+i
sin
(2kπ/n), k=0,1,2,…,n-1, i=-1, and let p=u+iv(v≠0) be a nonreal number. If p∈Ωn, or equivalently, p is a convex combination of the nth roots of unity; that is, p=λ0q0+λ1q1+⋯+λn-1qn-1, ∑k=0n-1λk=1, λk≥0, k=0,1,…,n-1; then there is a doubly stochastic matrix Cn such that
(34)p∈σ(Cn)={λ0+λ1qk+λ2qk2+⋯+λn-1qkn-1,k=0,1,…,n-1}.
Proof.
It is clear that the following permutation matrix:
(35)Pn=(01010⋱⋱⋱⋱110)
has spectrum σ(Pn)={q0,q1,…,qn-1} and then the following circulant matrix:
(36)Cn=(λ0λ1⋯λn-1λn-1λ0⋯λn-2⋮⋮⋱⋮λ1λ2⋯λ0)=λ0In+λ1Pn+⋯+λn-1Pnn-1
is doubly stochastic and has spectrum σ(Cn)={λ0+λ1qk+λ2qk2+⋯+λn-1qkn-1,k=0,1,…,n-1}. When k=1, we have λ0+λ1q1+λ2q12+⋯+λn-1q1n-1=p∈σ(Cn).
Theorem 12.
Let Λ={1,λ2=u+iv,λ3=u-iv} with u∈R and v>0. Then Λ can be realized by an IDS matrix if and only if
(37)-12≤u<1,v≤1-u3.
When (37) holds Λ to be realized by the irreducible doubly stochastic matrix,
(38)C3=(αβ1-α-β1-α-βαββ1-α-βα),
where
(39)α=1+2u3<1,β=1-u3(1-3v1-u).
Proof.
Assume that (37) holds. Then, in the complex plane, u+iv is inside the regular triangle whose vertices are all the 3rd roots of unit q0=1, q1=cos(2π/3)+i
sin
(2π/3)=-1/2+i(3/2), and q2=cos(4π/3)+isin(4π/3)=-1/2-i(3/2), and hence u+iv is a convex combination of q0,q1, and q2. It is not difficult to calculate
(40)u+iv=α+βq1+(1-α-β)q2,
where α,β is given by (39). Therefore the spectrum σ(C3) of the irreducible doubly stochastic matrix C3 given in (38) contains u+iv by Lemma 11. Since C3 is a doubly stochastic matrix, we have σ(C3)=Λ and hence the sufficiency is proved. To prove the necessity, assume that Λ is realized by a doubly stochastic matrix C=(crs),crs≥0, r,s=1,…,n. Then 1+2u=tcC≥0 from which follows u≥-1/2, and the sum of products of pairs eigenvalues of C is
(41)2u+u2+v2=∑1≤r<s≤3det(crrcrscsrcss)≤∑1≤r<s≤3crrcss≤13(∑r=13crr)2=13(tcC)2=13(1+2u)2,
from which follows 3v2≤(1-u)2. Therefore, (37) holds.
Remark 13.
The necessary and sufficient condition (37) for the 3×3 DSIEP was given by Theorems 12 and 14 of [9].
Using Theorems 2 and 12, we have the following corollaries.
Corollary 14.
Let Λ={1,λ2,λ3=u+iv,λ4=u-iv}, 1>λ2≥-1, u,v(>0)∈R, and c=(3+λ2)/4. If
(42)-12≤uc<1,vc≤1-u/c3,
then Λ can be realized by a 4×4 irreducible stochastic matrix.
Proof.
Let m=3, λ2=((m+1)λ+m-1)/2m=(4λ+2)/6; then λ=(3λ2-1)/2∈(-1,1), c=(2m-1+λ)/2m=(5+λ)/6=(3+λ2)/4>0. If (42) holds, then {1,(u+iv)/c,(u-iv)/c} is realized by the irreducible doubly stochastic matrix
(43)B=(αβ1-α-β1-α-βαββ1-α-βα)
by Theorem 12, where
(44)α=1+2u/c3,β=1-u/c3(1-3v/c1-u/c).
Now, the 4×4 matrix (given in (7)) as follows:
(45)C=(1+λ21-λ23u3T1-λ23u3cB)
is an irreducible doubly stochastic matrix b whom {1,c((u+iv)/c),c((u-iv)/c),λ2}=Λ is realized (Theorem 2).
Corollary 15.
Let Λ={1,λ2,λ3,u+iv,u-iv} be a complex 5-tuple with 1>λ2≥λ3≥-1, u2<u2+v2≤1,v>0, and c*=(4+λ3)/5, c′=(3+λ2/c*)/4, c=c*c′, then c*∈(11/15,1), c′>7/11,0<c<c′. If
(46)λ3≥-14,1>λ2c≥-1,(uc)2+(vc)2≤1,-12≤uc,vc≤1-u/c3,
then Λ is realized by a 5×5 irreducible stochastic matrix.
Proof.
Let m=4, λ3=((m+1)λ+m-1)/2m=(5λ+3)/8; then λ=(8λ3-3)/5∈(-1,1), c*=(2m-1+λ)/2m=(7+λ)/8=(4+λ3)/5∈(3/4,1) and hence c′≥(3-4/3)/4=5/12. If Λ satisfies Condition (46), then {1,λ2/c*,(u+iv)/c*,(u-iv)/c*} is realized by an irreducible 4×4 doubly stochastic matrix B* by Corollary 14. Now for λ=(8λ3-3)/5, the 5×5 matrix (given in (7))
(47)C=(1+λ21-λ4u4T1-λ4u4c*B*)
is an irreducible doubly stochastic by whom {1,c*(λ2/c*),c*((u+iv)/c*), c*((u-iv)/c*), λ3}=Λ is realized by Theorem 2.
Theorem 16.
Let Λ={λ1,λ2,λ3,λ4}={1,-2u-1,u+iv,λ3=u-iv}, u,v(>0)∈R contain a pair of conjugate complex numbers such that λ1+λ2+λ3+λ4=0. If
(48)-12≤u≤0,1+u-v≥0,
then Λ is realized by a 4×4 irreducible doubly stochastic matrix with zero trace.
Proof.
Assume that (48) holds. Then, in the complex plane, u+iv is inside the right triangle whose vertices are q1=i, q2=-1, q3=-i and hence u+iv is a convex combination of q1,q2, and q3. It is not difficult to calculate
(49)u+iv=1+u-v2q1+(-u)q2+1+u+v2q3.
Since q0=1, q1=i, q2=-1, q3=-i are all the 4th roots of unit, Lemma 11 asserts that the spectrum of the following 4×4 irreducible doubly stochastic matrix:
(50)(01+u-v2-u1+u+v21+u+v201+u-v2-u-u1+u+v201+u-v21+u-v2-u1+u+v20)
is {((1+u-v)/2)ik-u(-1)k+((1+u+v)/2)(-i)k, k=0,1,2,3}={1,u-iv,-1-2u, u+iv}=Λ.
Corollary 17.
Let Λ={λ1,λ2,λ3,λ4,λ5}={1,-1/4,(-3-8u)/4,u+iv,u-iv},u,v(>0)∈R and c=(3+λ2)/4=11/16. If
(51)-12≤uc≤0,1+uc-vc≥0,
then Λ is a realized by a 5×5 irreducible stochastic matrix.
Proof.
Let m=4, λ2=((m+1)λ+m-1)/2m=(5λ+3)/8; then λ=-1. If (51) holds, then {1,-1-2u/c,(u+iv)/c,(u-iv)/c} is realized by the irreducible doubly stochastic matrix(52)B=(01+u/c-v/c2-u/c1+u/c+v/c21+u/c+v/c201+u/c-v/c2-u/c-u/c1+u/c+v/c201+u/c-v/c21+u/c-v/c2-u/c1+u/c+v/c20)by Theorem 16. Now the following 5×5 matrix (given in (7)):
(53)C=(1+λ21-λ4u4T1-λ4u4cB)=(012u4T12u4cB)
is an irreducible doubly stochastic matrix with zero trace by whom {1,c((u+iv)/c),c((u-iv)/c),λ2,-1-λ2-2u}=Λ is realized by Theorem 2.
Theorem 18.
Let Λ={1,λ2=u+iv,λ3=u-iv,λ4=u′+iv′,λ5=u-iv′} contain two pairs of conjugate complex numbers with u, v(>0) given and u′, v′(>0) depending on u,v (i=-1). If
(54)cos4π5≤u<1,v<min{sin(2π/5)(u-1)cos(2π/5)-1,cos(4π/5)ffffff+(sin(2π/5)-sin(4π/5))(u-cos(2π/5))cos(2π/5)-cos(4π/5)},
then Λ is realized by a 5×5 irreducible doubly stochastic matrix, where u′, v′ are depending on u,v (see (58) and (63)).
Proof.
Assume that (54) holds. Then, in the complex plane, u+iv is inside the right pentagon whose vertices are all the 5th roots of units q0=1, qk=cos(2kπ/5)+i
sin
(2kπ/5), k=1,2,3,4 (see Figure 1) and hence u+iv is a convex combination of q0,q1,q2,q3,q4. There are two cases to be considered.
Case 1. u,v satisfy (54) and v≤sin(4π/5)(u-1)/(cos(4π/5)-1). In this case, u+iv is inside Triangle Δq0q2q3 and hence u+iv is a convex combination of q0,q2,q3. A calculation yields
(55)u+iv=λ0+λ2q2+λ3q3,
where
(56)λ0=u-cos(4π/5)1-cos(4π/5),λ2=1-u1-cos(4π/5)(12-(1-cos(4π/5))v2sin(4π/5)(1-u)),λ3=1-λ0-λ2.
Now the following irreducible circulant matrix that is also doubly stochastic:
(57)C5=(λ00λ2λ300λ00λ2λ3λ30λ00λ2λ2λ30λ000λ2λ30λ0)
has spectrum: σ(C5)={λ0+λ2qk2+λ3qk3,k=0,1,2,3,4} by Lemma 11. Taking k=0,1,4, we have 1,u±iv∈σ(C3) and taking k=2,3, we have that λ0+λ2q4+λ3q1 and λ0+λ2q1+λ3q4 are in σ(C3) and are conjugate to each other. Therefore, if we set
(58)u′=λ0+λ2cos8π5+λ3cos2π5,v′=|λ2sin8π5+λ3sin2π5|,
then Λ is realized by the 5×5 irreducible doubly stochastic matrix C5.
Case 2. u,v satisfy (54) and v>sin(4π/5)(u-1)/(cos(4π/5)-1). In this case, u+iv is inside Triangle Δq0q1q2 and hence u+iv is a convex combination of q0,q1,q2. A calculation yields
(59)u+iv=λ0′+λ1′q1+λ2′q2,
where
(60)λ0′=u-x1-x,λ1′=(x-cos(4π/5))(1-λ0′)cos(2π/5)-cos(4π/5),λ2′=1-λ0′-λ1′,
with
(61)x=((sin2π5-(sin(2π/5)-sin(4π/5)cos(2π/5)-cos(4π/5))×cos2π5((sin(2π/5)-sin(4π/5))(cos(2π/5)-cos(4π/5))))(u-1)+v)×((sin(2π/5)-sin(4π/5)cos(2π/5)-cos(4π/5))(1-u)+v)-1.
Now the irreducible circulant matrix that is also doubly stochastic as follows:
(62)C5′=(λ0′λ1′λ2′000λ0′λ1′λ2′000λ0′λ1′λ2′λ2′00λ0′λ1′λ1′λ2′00λ0′)
has spectrum σ(C5′)={λ0′+λ1′qk+λ2′qk2,k=0,1,2,3,4} by Lemma 11. Taking k=0,1,4, we have 1,u±iv∈σ(C3′) and taking k=2,3, we have that λ0′+λ1′q2+λ2′q4 and λ0′+λ1′q3+λ2′q1 are in σ(C3′) and are conjugate to each other. Therefore, if we set
(63)u′=λ0′+λ1′cos4π5+λ2′cos8π5,v′=|λ1′sin4π5+λ2′sin8π5|,
then Λ is realized by the 5×5 irreducible doubly stochastic matrix C5′.
Example 19.
Λ1={1,-0.3+0.6i,-0.3-0.6i} satisfies Condition (37) of Theorem 12 and is doubly stochastic realized by
(64)A1=(0.133330.086920.779740.779740.133330.086920.086920.779740.13333).
Example 20.
Let Λ2={1,λ2,u+vi,u-vi}={1,0.46,-0.3+0.6i,-0.3-0.6i}, c=(3+λ2)/4=0.865. Then Condition (42) of Corollary 14 is satisfied and Λ2 is doubly stochastic realized by
(65)A2=(0.5950.1350.1350.1350.1350.088330.041920.737440.1350.737440.088330.041920.1350.041920.737440.08833).
Example 21.
Let Λ3={1,λ2,λ3,u+vi,u-vi}={1,0.6,0.1,-0.3+0.6i,-0.3-0.6i}. Then c*=(4+λ3)/5=0.82, c′=(3+λ2/c*)/4=0.93295, c=c*c′=0.765, and Condition (46) of Corollary 15 is satisfied and Λ3 is doubly stochastic realized by
(66)A3=(0.280.180.180.180.180.180.6550.0550.0550.0550.180.0550.0550.008590.701410.180.0550.701410.0550.008590.180.0550.008590.701410.055).
Example 22.
Let Λ4={1,-1/4,(-3-8u)/3,u+iv,u-iv} with u=-0.2,v=0.4 and c=3/4. Then Condition (51) of Corollary 17 is satisfied and Λ4 is doubly stochastic realized by
(67)A4=(00.250.250.250.250.2500.4750.20.0750.250.07500.4750.20.250.20.07500.4750.250.4750.20.0750).
Example 23.
Let Λ5={1,u+vi,u-vi,u′+v′i,u′-v′i}, where u=0.3, v=0.6 and u′,v′(>0) will be determined later. It is easy to verify that u,v satisfy Condition (54) of Theorem 18 and v>sin(4π/5)(u-1)/(cos(4π/5)-1) (i.e., Λ4 belongs to Case 2) and u+vi=λ0+λ1q1+λ2q2=0.29613+0.51278(cos(2π/5)+sin(2π/5)i)+0.19109(cos(4π/5)+sin(4π/5)i). Let u′=λ0+λ1cos(2π/5)+λ2cos(4π/5)=-0.05966, v′=|λ1sin(2π/5)+λ2sin(4π/5)|=0.11967. Then Λ4={1,u+vi,u-vi,u′+v′i,u′-v′i} is doubly stochastic realized by the following irreducible doubly stochastic matrix:
(68)A5=(0.296130.512780.191090000.296130.512780.191090000.296130.512780.191090.19109000.296130.512780.512780.19109000.29613).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors thank very much the anonymous referee whose valuable comments and suggestions helped them to improve the representation of the paper. The first author’s work is supported by The Doctorate Point Foundation of the Educational Ministry of China (no. 20113401130001) and the National Natural Science Foundation of China-Guangdong Joint Found (no. U1201255). Changqing 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)
BorobiaA.On the nonnegative eigenvalue problem1995223/22413114010.1016/0024-3795(94)00343-CMR1340689ZBL0831.15014FangM.A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices2010432112925292710.1016/j.laa.2009.12.032MR2639254ZBL1193.15009HwangS.-G.PyoS.-S.The inverse eigenvalue problem for symmetric doubly stochastic matrices2004379778310.1016/S0024-3795(03)00366-5MR2039298ZBL1040.15010KaddouraI.MouradB.On a conjecture concerning the inverse eigenvalue problem of 4×4 symmetric doubly stochastic matrices200831315131519MR2447643ZBL1166.15004LaffeyT. J.MeehanE.A characterization of trace zero nonnegative 5×5 matrices1999302/30329530210.1016/S0024-3795(99)00099-3MR1733536ZBL0946.15008LoewyR.LondonD.A note on an inverse problem for nonnegative matrices1978618390MR048056310.1080/03081087808817226MouradB.On a spectral property of doubly stochastic matrices and its application to their inverse eigenvalue problem201243693400341210.1016/j.laa.2011.11.034MR2900724ZBL1247.15030NazariA. M.SherafatF.On the inverse eigenvalue problem for nonnegative matrices of order two to five201243671771179010.1016/j.laa.2011.12.023MR2889958ZBL1241.15008PerfectH.MirskyL.Spectral properties of doubly-stochastic matrices1965693557MR0175917ZBL0142.00302ReamsR.An inequality for nonnegative matrices and the inverse eigenvalue problem199641436737510.1080/03081089608818485MR1481909ZBL0887.15015RojoO.SotoR. L.Existence and construction of nonnegative matrices with complex spectrum2003368536910.1016/S0024-3795(02)00650-XMR1983194ZBL1031.15017Torre-MayoJ.Abril-RaymundoM. R.Alarcia-EstévezE.MarijuánC.PisoneroM.The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial. EBL digraphs20074262-372977310.1016/j.laa.2007.06.014MR2350690ZBL1136.15007YangS.LiX.Inverse eigenvalue problems of 4×4 irreducible nonnegative matrices2008IWorld Academic UnionYangS.XuC.Row stochastic inverse eigenvalue problem20112011article 24510.1186/1029-242X-2011-24MR2823624ZBL1269.15008HornR. A.JohnsonC. R.1985Cambridge, UKCambridge University PressMR832183