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Nonnegative matrices with prescribed spectrum and elementary divisors
Linear Algebra and its Applications 439 (2013) 3591–3604
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Linear Algebra and its Applications www.elsevier.com/locate/laa
Nonnegative matrices with prescribed spectrum and elementary divisors ✩ Ricardo L. Soto a,∗ , Roberto C. Díaz a , Hans Nina a,b , Mario Salas a a b
Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile Departamento de Matemáticas, Universidad Mayor de San Andrés, La Paz, Bolivia
a r t i c l e
i n f o
Article history: Received 20 June 2013 Accepted 21 September 2013 Available online 16 October 2013 Submitted by R. Brualdi MSC: 15A29 15A18
a b s t r a c t In this paper we give new sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors, which drastically improve and contain some of the previous known conditions. We also show how to perturb complex eigenvalues of a nonnegative matrix while keeping its nonnegativity. These results allow us, under certain conditions, to easily decide if a given list is realizable with prescribed elementary divisors. © 2013 Elsevier Inc. All rights reserved.
Keywords: Nonnegative inverse elementary divisors problem Nonnegative matrices
1. Introduction Let A ∈ Cn×n and let
⎡J n1 (λ1 ) ⎢ J ( A ) = S −1 A S = ⎢ ⎣
⎤ ⎥ ⎥ ⎦
J n2 (λ2 )
..
. J nk (λk )
be the Jordan canonical form of A (hereafter JCF of A). The ni × ni submatrices ✩
*
Supported by Fondecyt 1120180, Chile, Mecesup UCN0711. Corresponding author. Fax: +56 55 355599. E-mail addresses: [email protected] (R.L. Soto), [email protected] (R.C. Díaz), [email protected] (H. Nina), [email protected] (M. Salas).
0024-3795/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.09.034
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⎡λ
i
⎢ ⎢ ⎣
λi
J ni (λi ) = ⎢
⎤
1
..
.
..
.
⎥ ⎥ ⎥, ⎦
1
i = 1, 2, . . . , k
λi are called the Jordan blocks of J ( A ). Then the elementary divisors of A are the polynomials (λ − λi )ni , that is, the characteristic polynomials of J ni (λi ), i = 1, . . . , k. The nonnegative inverse elementary divisors problem (NIEDP) is the problem of determining necessary and sufficient conditions under which the polynomials
(λ − λ1 )n1 , (λ − λ2 )n2 , . . . , (λ − λk )nk ,
n1 + · · · + nk = n,
are the elementary divisors of an n × n entrywise nonnegative matrix A [7,12]. The NIEDP contains the nonnegative inverse eigenvalue problem (NIEP), which asks for necessary and sufficient conditions for the existence of an entrywise nonnegative matrix with prescribed spectrum. Both problems remain unsolved. If Λ = {λ1 , λ2 , . . . , λn } is the spectrum of a nonnegative matrix A, we say that Λ is realizable and that A is the realizing matrix. A matrix A = (ai j )ni, j =1 is said to have constant row sums if all its rows sum up to the same constant, say
α , i.e.,
n
j =1 ai j
= α , i = 1, . . . , n. The set of all matrices with constant row sums equal
to α is denoted by CS α . It is clear that any matrix in CS α has the eigenvector e = (1, 1, . . . , 1) T corresponding to the eigenvalue α . Denote by ek the vector with one in the k-th position and zeros elsewhere. The relevance of matrices with constant row sums is due to the well-known fact that if Λ = {λ1 , λ2 , . . . , λn } is the spectrum of an n × n nonnegative matrix, then Λ is also the spectrum of a nonnegative matrix with constant row sums equal to its Perron eigenvalue. Let S be a nonsingular matrix such that S −1 A S = J ( A ) is the JCF of A. If A ∈ CS λ1 , then S can be chosen such that Se1 = e and the rows of S −1 = ( si , j ) satisfy n
s1, j = 1 and
n
s i , j = 0,
j =1
j =1
i = 2, . . . , n .
If T and S are n × n matrices of the form
⎡ ⎢
⎤ ··· ∗ . . .. ⎥ . .⎥ ∗ ⎥ .. .. ⎥ . . ∗⎦ .. . ∗ ∗ ∗
λ1
⎢0 T =⎢ ⎢ .. ⎣ . 0
⎡
1
s12
⎢1 S =⎢ ⎢ .. ⎣.
s22
1
sn2
⎢
and
with S being nonsingular, then S T S −1 e
..
.
⎤ · · · s1n .. . s2n ⎥ ⎥ ⎥ .. ⎥ , .. . . ⎦ .. . s nn
λ1 e, that is, S T S −1
= ∈ CS λ1 . We shall denote by σ ( A ) the spectrum of A, by E i j the n × n matrix with 1 in the (i , j ) position and zeros elsewhere, by rank( X ) the rank of the matrix X , and Im λ will be taken as nonnegative. Let E be the matrix E =±
E i , i +1 ,
K ⊂ {1, 2, . . . , n − 1}
i∈ K
and let A be an n × n complex matrix with JCF J ( A ) = S −1 A S. Then for an appropriate set K
J ( A ) + E = S −1 A S + E
= S −1 A + S E S −1 S
is the JCF of A + S E S −1 . Thus, given a complex matrix A with J ( A ) = S −1 A S, we may obtain, trivially, a matrix B = A + S E S −1 with σ ( B ) = σ ( A ) and prescribed elementary divisors. Moreover, if A ∈ CS λ1 and S = [e | ∗ | · · · | ∗], then ( A + S E S −1 ) ∈ CS λ1 .
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The following two results, which we set here for completeness, have been shown to be very useful, not only to derive sufficient conditions for the realizability of both problems, the NIEP and the NIEDP, but for constructing a realizing matrix as well. The first result, due to Brauer [2], shows how to modify one single eigenvalue of a matrix, via a rank-one perturbation, without changing any of its remaining eigenvalues (see [10,1,12] and the references therein to see how Brauer’s result has been applied to NIEP and NIEDP). The second result, due to Rado and introduced by Perfect in [9] is an extension of Brauer’s result and shows how to change r eigenvalues of a matrix (r < n), via a perturbation of rank r, without changing any of its remaining n − r eigenvalues (see [9,11,13] to see how Rado’s result has been applied to NIEP). Theorem 1.1 (Brauer). (See [2].) Let A be an n × n arbitrary matrix with eigenvalues λ1 , λ2 , . . . , λn . Let v = ( v 1 , . . . , v n )T be an eigenvector of A corresponding to λk and let q be any n-dimensional vector. Then the matrix A + vq T has eigenvalues λ1 , . . . , λk−1 , λk + v T q, λk+1 , . . . , λn . Theorem 1.2 (Rado). (See [9].) Let A be an n × n arbitrary matrix with spectrum Λ = {λ1 , . . . , λn }. Let X = [x1 | · · · | xr ] be such that rank( X ) = r and Axi = λi xi , i = 1, . . . , r, r n. Let C be an r × n arbitrary matrix. Then A + X C has eigenvalues μ1 , . . . , μr , λr +1 , . . . , λn , where μ1 , . . . , μr are eigenvalues of the matrix Ω + C X with Ω = diag{λ1 , . . . , λr }. Next we recall the following results due to Laffey and Šmigoc [6], which will be used later: Theorem 1.3. (See [6].) Let Λ = {λ1 , λ2 , . . . , λn } be a list of complex numbers with Λ = Λ, Re λk 0, k = 2, . . . , n. Then Λ is the nonzero spectrum of a nonnegative matrix if
i)
s1 =
n
λk 0,
k =1
ii)
s2 =
n
λk2 > 0.
k =1
The minimal number N of zeros that need to be added to Λ to make it realizable satisfies s21 (n + N )s2 . Furthermore, the list Λ = {λ1 , . . . , λn , 0, . . . , 0} can be realized by a nonnegative matrix of the form C + α I , where C is a companion matrix with trace zero, α 0 and I is the identity matrix of the appropriate size. Theorem 1.4. (See [6].) Let Λ = {λ1 , λ2 , . . . , λn } be a list of complex numbers with Λ = Λ, Re λk 0, k = 2, . . . , n. Then Λ is realizable if and only if
i)
s1 =
n
λk 0,
k =1
ii)
s2 =
n
λk2 0,
k =1
iii)
s21 ns2 .
In [7] Minc proves that given an n × n diagonalizable positive matrix A, then there exists an n × n positive matrix B, with σ ( B ) = σ ( A ) and with arbitrarily prescribed elementary divisors, provided that elementary divisors corresponding to nonreal eigenvalues occur in conjugate pairs. As a consequence of this result and results from the NIEP for symmetric matrices, the authors in [12] give sufficient conditions for the existence and construction of an n × n positive matrix with prescribed spectrum and prescribed elementary divisors. The following result was given in [12]:
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Lemma 1.1. Let q be an arbitrary n-dimensional vector and let A ∈ CS λ1 with JCF J ( A ) = S −1 A S. Let λ1 + n q = λi , i = 2, . . . , n. Then A + eqT has JCF i i =1
J ( A) +
n
qi E 11 .
i =1
In particular, if
n
i =1 q i
= 0, then A and A + eqT are similar.
In [12] the authors completely solve the NIEDP in two particular cases: for lists of real numbers satisfying λ1 > λ2 · · · λn 0 and for Suleimanova lists λ1 > 0 λ2 · · · λn . For the general case, the authors give some sufficient conditions. The proofs are constructive, in the sense that if the conditions are satisfied then we can always construct a solution matrix. The paper is organized as follows: In Section 2, based on results of Brauer and Rado, we give new sufficient conditions for the NIEDP to have a solution, which drastically improve and contain known previous conditions in [12]. We prove, in Section 3, results which show how to perturb complex eigenvalues of a nonnegative matrix while keeping its nonnegativity. These results allow us, under certain conditions, to easily decide the realizability of a given list with prescribed elementary divisors. Finally, in Section 4, we show some examples to illustrate the results. 2. New conditions for the NIEDP We start this section with a general result, which gives a very simple sufficient condition for the existence and construction of a nonnegative matrix with prescribed spectrum and elementary divisors. This result contains Theorems 2, 3 and 4 in [12] as particular cases. Theorem 2.1. Let Λ = {λ1 , λ2 , . . . , λn } be a list of complex numbers with Λ = Λ, i = 2, . . . , n. Let
M = max {0, Re λi + Im λi }; 2i n
m=−
n
min{0, Re λi , Im λi }.
n
i =1 λi
0, and λ1 |λi |,
(1)
i =2
If
i ) λ1 M + m ,
(2)
when all possible Jordan blocks J ni (λi ), of size ni 2, are associated to a real eigenvalue λi < 0; or when there is at least one Jordan block J ni (λi ), of size ni 2, associated to a real eigenvalue λi 0 with M = Re λi 0 + Im λi 0 > |λk |, 2 k p, for some i 0 , p + 1 i 0 n − 1, or if
ii)
λ1 > M + m ,
(3)
when at least one Jordan block J ni (λi ), of size ni 2, is associated to a real eigenvalue λi 0 with M = λk 0, for some k, 2 k p, then there exists an n × n nonnegative matrix A ∈ CS λ1 with spectrum Λ and with prescribed elementary divisors
(λ − λ1 ), (λ − λ2 )n2 , . . . , (λ − λk )nk ,
n2 + · · · + nk = n − 1.
Proof. Let λ1 > λ2 · · · λ p be real numbers and let
λ p +1 , λ p +1 , . . . , λn−1 , λn−1 = λn be complex nonreal numbers. Let
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⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ J =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
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⎤
λ1 ..
. λp Re λ p +1
Im λ p +1
−Im λ p +1
Re λ p +1
..
. Re λn−1 −Im λn−1
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ Im λn−1 ⎦ Re λn−1
It is clear that the JCF of J is D = diag{λ1 , λ2 , . . . , λn }. Let E = i ∈ K E i ,i +1 , K ⊂ {2, . . . , n − 1} such that D + E is the prescribed JCF. Let S = [e | e2 | · · · | en ]. Then S E S −1 = i ∈ K (− E i ,1 + E i ,i +1 ) and B = S ( J + E ) S −1 is a matrix in CS λ1 with spectrum Λ and with the prescribed JCF. Now let q T = (q1 , q2 , . . . , qn ), where
q1 = −m, qk = −min{0, Re λk , −Im λk },
k = 2, 3, . . . , n .
We shall prove that A = B + eq is nonnegative. Let us first consider case i): If appears in at least one position (i , i + 1), i = 2, . . . , n − 1, then it is clear for < 0 with | | min{min2k p |λk |, min p +1kn Im λk }, that the entries of A in columns 2, 3, . . . , n, are all nonnegative. On the first column, the entries of A in positions k = 2, . . . , p are of the form λ1 − λk − m, or λ1 − λk − − m. Then they are all nonnegative if we choose < 0 with | | min2k p |λk |, whenever appears in at least one position (i , i + 1), i = 2, . . . , p. In positions k = p + 1, . . . , n, the entries on the first column of A are of the form λ1 − (Re λi + Im λi ) − m or λ1 − (Re λi − Im λi ) − m − , which from (2) are all nonnegative if we choose < 0, | | min p +1kn Im λk . If we have at least one Jordan block J ni (λi ), of size ni 2, associated to a real eigenvalue λi 0 and M occurs in a complex eigenvalue as M = Re λi 0 + Im λi 0 > |λk |, 2 k p, then we need that λ1 − λk − m. Hence we choose 0 < min2k p {λ1 − λk − m}. Then, in this case if we choose T
0 < min
min {λ1 − λk − m}, min |λk |,
2k p
2k p
min
p +1kn
Im λk ,
all entries in the first column of A are nonnegative. Now we consider case ii): If we choose
0 < min λ1 − ( M + m), min |λk |, 2k p
min
p +1kn
Im λk ,
the entries of A in columns 2, 3, . . . , n, are all nonnegative (for appearing in at least one position (i , i + 1)). In the first column, the entries of A in positions k = 2, . . . , p, are all nonnegative if we choose 0 < min{λ1 − ( M + m), min2k p |λk |}. In positions k = p + 1, . . . , n, if we choose 0 < min p +1kn Im λk the entries are all nonnegative. Then by choosing < 0 with | | min{min2k p |λk |, min p +1kn Im λk } or
0 < min
min {λ1 − λk − m}, min |λk |,
2k p
2k p
min
p +1kn
Im λk ,
in the case i) or by choosing
0 < min λ1 − ( M + m), min |λk |, 2k p
min
p +1kn
Im λk
in the case ii), all entries in the first column of A are nonnegative. It is clear from (2), that if does T not appear n in at least one position (i , i + 1), i = 2, . . . , n − 1, A = B + eq is still nonnegative. Finally, q = 0, from Lemma 1.1 A has the prescribed elementary divisors. 2 since i i =1
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The following corollary improves Theorem 4 in [12] and completely solves the NIEDP for complex lists Λ = {λ1 , λ2 , . . . , λn } with Re λi < 0, |Re λi | Im λi , i = 2, . . . , n. Corollary 2.1. Let Λ = {λ1 , λ2 , . . . , λn } be a list of complex numbers with Re λi < 0, |Re λi | Im λi , i = 2, . . . , n. Then there exists a nonnegative matrix A ∈ CS λ1 with spectrum Λ and with prescribed elementary divisors
(λ − λ1 ), (λ − λ2 )n2 , . . . , (λ − λk )nk ,
n2 + · · · + nk = n − 1
if and only if λ1 + λ2 + · · · + λn 0. Proof. The condition is necessary for the existence of a nonnegative matrix with spectrum Λ. From Theorem 2.1 the condition is sufficient. In fact,
M = max {0, Re λi + Im λi } = 0, 2i n
m=−
n
min{0, Re λi , Im λi } = −
i =2
n
Re λi = −
i =2
Then λ1 − m 0, that is, λ1 + λ2 + · · · + λn 0.
n
λi .
i =2
2
Remark 2.2. Remember that Rado’s Theorem 1.2 allows us to change r eigenvalues of a matrix A without changing any of its remaining n − r eigenvalues. When we apply Rado’s Theorem to obtain a nonnegative matrix A + X C with spectrum Λ = {λ1 , . . . , λn }, we start with a block diagonal matrix A where each block A k , k = 1, . . . , r, is nonnegative with Perron eigenvalue ωk and spectrum Γk = {ωk , λk2 , . . . , λkpk }. Then A + X C will be nonnegative with new eigenvalues λ1 , . . . , λr instead of ω1 , . . . , ωr . The r columns xi of X are linearly independent eigenvectors of A corresponding to the eigenvalues ωi and C is such that Ω + C X , with Ω = diag{ω1 , . . . , ωr }, has the new eigenvalues λ1 , . . . , λr . Thus, A + X C will have the desired spectrum Λ. The following result shows that the diagonal blocks A 1 , . . . , A r can have negative entries, which we may take away by setting appropriate entries in the suitable positions of the matrix C . Lemma 2.1. Let A be an n × n real block diagonal matrix, where each diagonal block A k ∈ CS λk1 (not necessarily nonnegative) has spectrum
Λk = {λk1 , λk2 , . . . , λkpk },
k = 1, 2, . . . , r ,
satisfying conditions of Theorem 2.1. Let qkT = (qk1 , . . . , qkpk ) with negative. Then M = A + X C is nonnegative with spectrum
pk
q j =1 kj
= 0, such that A k + e qkT is non-
{μ1 , μ2 , . . . , μr , λ12 , . . . , λ1p 1 , λ22 , . . . , λ2p 2 , . . . , λr2 , . . . , λrpr }, where μ1 , . . . , μr are eigenvalues of Ω + C X with Ω = diag{λ11 , λ21 , . . . , λr1 }. Proof. Let A = diag{ A 1 , A 2 , . . . , A r } and Axk = λk1 xk , k = 1, . . . , r, where xk has pk ones in positions p (k−1) + 1 to pk , with p 0 = 0, and zeros elsewhere. Let qk be an n-dimensional vector containing the qkT = (qk1 , qk2 , . . . , qkpk ) in positions p (k−1) + 1 to pk , with p 0 = 0 and zeros elsewhere. That is vector
xk = (0, . . . , 0, 1, . . . , 1, 0, . . . , 0) T ,
pk
qkT
= (0, . . . , 0, qk1 , . . . , qkpk , 0, . . . , 0). pk
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3597
Let
X = [x1 | x2 | · · · | xr ]
and
C B = q1T q2T · · · qrT .
Then
M = A + X C B = x1 q1T + x2 q2T + · · · + xr qrT r is nonnegative with spectrum k=1 Λk . Observe that M is block diagonal with each diagonal block being nonnegative. Now, to change the eigenvalues λ11 , λ21 , . . . , λr1 to μ1 , μ2 , . . . , μr , we apply Rado’s Theorem to obtain a nonnegative matrix
M + X C R = ( A + X C B ) + X C R = A + X (C B + C R ) = A + X C , where C = C B + C R , with the prescribed spectrum.
2
Let A, X , C , and Ω be as in Theorem 1.2 and let S be nonsingular and partitioned as S = [ X | Y ] with S −1 = [ U ]. Then U X = I r , V Y = I n−r , V X = U Y = 0 and A X = X Ω . Hence we have V
S
−1
AS =
U V
S −1 X C S =
A[ X | Y ] = CX 0
and
S −1 ( A + X C ) S =
CY 0
U AY V AY
Ω 0
(4)
,
,
Ω +CX 0
C Y + U AY V AY
(5)
,
where B = Ω + C X is an r × r matrix with eigenvalues μ1 , . . . , μr (the new eigenvalues) and diagonal entries λ1 , . . . , λr (the former eigenvalues). Then from a result in [8, Chapter VI, Lemma 1.2], if Ω and V AY in (4) have no eigenvalues in common, A is similar to Ω ⊕ V AY . In the same way, if B = Ω + C X and V AY in (5) have no common eigenvalues, A + X C is similar to B ⊕ V AY . Then we have: Lemma 2.2. Let A, X , Y , V , C , and Ω be as above. If the matrices B = Ω + C X and V AY have no common eigenvalues, then
J ( A + X C ) = J ( B ) ⊕ J ( V AY ). In particular, if C X = 0, A and A + X C are similar. Now we extend Theorem 2.1: Theorem 2.3. Let Λ = {λ1 , λ2 , . . . , λn } be a list of complex numbers with Λ = Λ, λ1 maxi |λi |, i = 2, . . . , n; n λ 0. Let Λ = Λ1 ∪ · · · ∪ Λ p 1 +1 be a pairwise disjoint partition, with Λk = {λk1 , λk2 , . . . , λkpk }; i i =1 λ11 = λ1 , k = 1, . . . , p 1 + 1, where Λ1 is realizable, p 1 is the number of elements of the list Λ1 and some lists Λk can be empty. Let ω2 , . . . , ω p 1 +1 be real numbers satisfying 0 ωk λ1 , k = 2, . . . , p 1 + 1. Suppose that i) for each k = 2, . . . , p 1 + 1, there exists a list Γk = {ωk , λk1 , . . . , λkpk } with M k + mk , as in Theorem 2.1, where
M k = max {0, Re λki + Im λki }; 1 i p k
mk = −
pk
ωk Mk + mk or ωk >
min{0, Re λki , Im λki },
i =1
and ii) there exists a p 1 × p 1 nonnegative matrix B with spectrum Λ1 and diagonal entries ω2 , . . . , ω p 1 +1 .
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Then there exists an n × n nonnegative matrix M ∈ CS λ1 with spectrum Λ and with prescribed elementary divisors
(λ − λ1 ), (λ − λ2 )n2 , . . . , (λ − λk )nk , Proof. Let
⎡
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Jk = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
n2 + · · · + nk = n − 1.
⎤
ωk ..
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ −1 )
. λks Re λk(s+1) Im λk(s+1)
−Im λk(s+1) Re λk(s+1) ..
. Re λk( pk −1) Im λk( pk −1)
−Im λk( pk Re λk( pk −1)
with spectrum Γk and let E k = E i ,i +1 such that J k + E k is the prescribed JCF. Let S k = [e | e2 | · · · | e pk ]. Then A k = S k ( J k + E k ) S k−1 ∈ CS ωk , k = 2, . . . , p 1 + 1 and
⎡
⎤
A2
⎢
⎥ ⎥. ⎦
A3
A=⎢ ⎣
..
. A p 1 +1
Observe from Lemma 2.1 that for each k = 2, . . . , p 1 + 1, we have that A k + ek qkT is nonnegative for
p
k an appropriate vector qkT = (qk0 , qkp 1 , . . . , qkpk ) with q = 0 and from Lemma 1.1, A k + ek qkT j =0 kj has the prescribed elementary divisors. Then, we may apply Rado’s perturbation as in Lemma 2.1. Thus A + X C (with C = C B + C R ) is nonnegative with spectrum Λ and from Lemma 2.2 A has the prescribed elementary divisors. 2
Let us consider the following example: Example 2.1. Let Λ = {7, 1, −2, −2, −1 + 3i , −1 − 3i }. We construct a nonnegative matrix A with spectrum Λ and with elementary divisors
(λ − 7), (λ − 1), (λ + 2)2 , (λ + 1 − 3i ), (λ + 1 + 3i ). Consider the partition Λ = Λ1 ∪ Λ2 ∪ Λ3 ∪ Λ4 with
Λ1 = {7, −1 + 3i , −1 − 3i }; Γ2 = {4, −2, −2}; The matrix
⎡
4
B = ⎣ 34 7 0
0 1 7
3 8 7
Λ2 = {−2, −2};
Γ3 = {1, 1};
Λ3 = {1};
Λ4 = ∅ and
Γ4 = {0}.
⎤ ⎦
0
has the spectrum Λ1 with diagonal entries 4, 1, 0. Then from Theorem 2.3 we have
⎡
4 0 ⎢ 7 −2 ⎢ ⎢6 0
A=⎢
⎢ ⎣
0 −1 −2
⎤
⎡
1 ⎥ ⎢1 ⎥ ⎢ ⎥ ⎢1 ⎥+⎢ 1 0 ⎥ ⎢0 ⎦ ⎣0 0 1 0 0
⎤
0 0 0 0⎥⎡ ⎥ −4 0 0 ⎥ ⎣ 34 ⎥ 7 1 0⎥ ⎦ 0 1 0 0 1
2 0 0
2 0 0 0 0 0 0 7 0
3 8 7
0
⎤ ⎦
R.L. Soto et al. / Linear Algebra and its Applications 439 (2013) 3591–3604
⎡
0 ⎢ 3 ⎢ ⎢ 2
=⎢ ⎢ 34 ⎢ 7 ⎣ 34 7
0
2 2 0 1 2 0 0 0 0 0
0 0 0 1
0 0 0 0
0 0 1 0 7 0
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⎤
3 3⎥ ⎥ 3⎥ 8 7 8 7
⎥ ⎥ ⎥ ⎦
0
with the prescribed spectrum and elementary divisors. Corollary 2.2. Let Λ = {λ1 , . . . , λ p , λ p +1 , . . . , λs , λs+1 , . . . , λn } be a list of complex numbers with Λ = Λ, λ1 maxi |λi |, i = 2, . . . , n, ni=1 λi 0, where λ1 , . . . , λ p are real numbers, λ p +1 , . . . , λn are complex nonreal with Re λi 0, |Re λi | |Im λi |, for i = p + 1, . . . , s and Re λi 0 for i = s + 1, . . . , n. Let Λ = Λ1 ∪ Λ2 be a partition, where
Λ1 = {λ1 , λs+1 , . . . , λn }, Λ2 = {λ2 , . . . , λ p , λ p +1 , . . . , λs }, with Λ1 being realizable by a nonnegative companion matrix. Then Λ is realizable by a nonnegative matrix with any prescribed elementary divisors associated to the eigenvalues in Λ2 . Proof. From the hypothesis, there exists an (n − s + 1) × (n − s + 1) nonnegative companion matrix B with spectrum Λ1 and a diagonal element μ1 = b11 = tr B. It is clear from Corollary 2.1 that Γ2 = {μ1 , λ2 , . . . , λ p , λ p +1 , . . . , λs } is realizable by a nonnegative matrix A 2 with any prescribed elementary divisors. Let Γk = {0} for k = 3, . . . , n − s + 2. Then
⎡ ⎢
⎤
A2
A=⎢ ⎣
⎥ ⎥ ⎦
0
..
. 0
is nonnegative with spectrum
{μ1 , λ2 , . . . , λs , 0, . . . , 0 }. n−s
Now we apply Rado’s Theorem 1.2 to change μ1 , 0, . . . , 0 (n − s zeros) to λ1 , λs+1 , . . . , λn , and to obtain a nonnegative matrix M = A + X C with spectrum Λ and with the prescribed elementary divisors associated to the eigenvalues of Λ2 . 2 3. Complex eigenvalues perturbation Let Λ = {λ1 , λ2 , . . . , λn }, with λ2 ∈ R, be a list of complex numbers, which is realizable by a nonnegative matrix. Guo [3] proved that for all t > 0, the perturbed list Λt = {λ1 + t , λ2 ± t , λ3 , . . . , λn } is also realizable by a nonnegative matrix. The problem of perturbing the real parts of a pair of complex eigenvalues of a realizable list, by keeping the realizability of the new list, was considered by Laffey in [5] and by Guo and Guo in [4]. The following result shows that we can also perturb the imaginary parts of a pair of complex eigenvalues a + bi, a − bi of a realizable list, and to obtain a new list which is also realizable. Theorem 3.1. Let Λ = {λ1 , λ2 , . . . , λn } be a realizable list of complex numbers with Re λk 0, |Re λk | Im λk , k = 2, . . . , n. Let x p = Re λ p and y p = Im λ p , 2 p n − 1. Then for all t > 0, the perturbed lists
Λt+ = λ1 + 2t , λ2 , . . . , λ p −1 , x p + ( y p + t )i , x p − ( y p + t )i , λ p +2 , . . . , λn , Λt− = λ1 + 2t , λ2 , . . . , λ p −1 , x p + ( y p − t )i , x p − ( y p − t )i , λ p +2 , . . . , λn
are also realizable.
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Proof. Let λ1 , λ2 , . . . , λ p −1 be real numbers and let λ p , λ p , . . . , λn−1 , λn−1 be complex. Let us consider the matrix
⎡
⎤
λ1
⎢ ⎢ ⎢ ⎢ λ1 − λ p −1 ⎢ ⎢ λ1 − x p − y p − t Bt = ⎢ ⎢ λ1 − x p + y p + t ⎢ ⎢ ⎢ ⎢ ⎣λ − x −y 1
n −1
..
. λ p −1 xp
−( y p + t )
yp + t xp
..
n −1
λ1 − xn−1 + yn−1
Since Λ is realizable,
q1 =
n
n
i =1 λi
. xn−1
− y n −1
y n −1 xn−1
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
0. Let q T = (q1 , q2 , . . . , qn ), where
λi + t ,
i =1
qi = −λi ,
i = 2, . . . , p − 1,
q p = −x p + t , q j = −x j ,
q p +1 = − x p ,
j = p + 2, . . . , n .
Observe that B t ∈ CS λ1 , and since Re λk 0, |Re λk | Im λk , A = B t + eq T is nonnegative with spectrum
Λt+ = λ1 + 2t , λ2 , . . . , λ p −1 , x p + ( y p + t )i , x p − ( y p + t )i , λ p +2 , . . . , λn . It is clear that changing t to −t in the matrix B t and choosing q p = −x p and q p +1 = −x p + t, we obtain
Λt− = λ1 + 2t , λ2 , . . . , λ p −1 , x p + ( y p − t )i , x p − ( y p − t )i , λ p +2 , . . . , λn , which is also realizable.
2
Theorem 3.1 may be useful to decide the realizability of a list of complex numbers. Consider for example the list
Λ = {16, −2 + 5i , −2 − 5i , −3 + i , −3 − i }. It is clear from Corollary 2.1 (see also [1,12]) that the complex Suleimanova list
Λ = {6, −2 + 2i , −2 − 2i , −1 + i , −1 − i } is realizable. It is also realizable for example, by a nonnegative companion matrix. Then from Theorem 3.1 we have that the lists
Λ = {10, −2 + 2i , −2 − 2i , −3 + i , −3 − i } with t = 2, Λ = Λ = {16, −2 + 5i , −2 − 5i , −3 + i , −3 − i } with t = 3 are also realizable. We observe however, that with this procedure, the increase in the Perron eigenvalue may be greater than necessary. In fact, the Perron eigenvalue 16, in this example, may be replaced by λ1 10 since for λ1 = 10, the companion matrix with spectrum Λ is nonnegative. It is clear that we can also perturb the real parts of a pair of complex eigenvalues. In this case we have: Corollary 3.1. Let Λ = {λ1 , λ2 , . . . , λn } be a realizable list of complex numbers with Re λk 0, |Re λk | Im λk , k = 2, . . . , n. Let x p = Re λ p and y p = Im λ p , 2 p n − 1. Then for all t > 0, the perturbed lists
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Λt+ = {λ1 + t , λ2 , . . . , λ p −1 , x p + t + iy p , x p + t − iy p , λ p +2 , . . . , λn } and
Λt− = {λ1 + 2t , λ2 , . . . , λ p −1 , x p − t + iy p , x p − t − iy p , λ p +2 , . . . , λn } are also realizable. For lists satisfying Theorem 1.4 we have: Theorem 3.2. Let Λ = {λ1 , λ2 , . . . , λn } be a list of complex numbers with Λ = Λ, Re λk 0, Im λk 0. If Λ is realizable, that is, if Λ satisfies conditions of Theorem 1.4, then for all t > 0,
Λt− = λ1 + 2t , λ2 , . . . , λ p −1 , x p + i ( y p − t ), x p − i ( y p − t ), λ p +2 , . . . , λn and
Λt+ = λ1 + 4t , λ2 , . . . , λ p −1 , x p + i ( y p + t ), x p − i ( y p + t ), λ p +2 , . . . , λn are also realizable (Λt− and Λt+ satisfy Theorem 1.4). Proof. Let x p = Re λ p and y p = Im λ p , 2 p n − 1. First we prove that Λt− is realizable. Denote by s1 (t ) the sum of the elements of the list Λt− and by s2 (t ) the sum of the squares of the elements of Λt− . Then
s1 (t ) = s1 + 2t , s2 (t ) = s2 + 4(λ1 + y p )t + 2t 2 . It is clear that Λt− = Λt− . Since s1 0, s2 0 and λ1 + y p 0, then s1 (t ) 0 and s2 (t ) 0. It remains to show that s21 (t ) ns2 (t ). Observe that
s21 (t ) = s21 + 4s1 t + 4t 2 , ns2 (t ) = ns2 + 4n(λ1 + y p )t + 2nt 2 . Then s21 ns2 , 4 < 4n, s1 (λ1 + y p ) (as s1 λ1 and y p 0) and 4 2n (as n 3). Hence s21 (t ) ns2 (t ). In the same way we may prove that Λt+ is realizable. 2 Remark 3.3. It is clear that in the previous theorems we can perturb simultaneously two or more pairs of conjugate complex numbers, under the condition that we appropriately increase λ1 as well. It is also clear that we may perturb a pair of real numbers λk , λk to obtain λk + ti, λk − ti. The results in this section give rise to an interesting, useful and easy procedure to construct, from a realizable list with certain elementary divisors, a new realizable list, where the elementary divisors associated to the new eigenvalues keep the structure of elementary divisors associated to the former eigenvalues. Theorem 3.4. Let Λ = {λ1 , λ2 , . . . , λn } be a list of complex numbers with Λ = Λ, λ1 maxi |λi |, i = 2, . . . , n, n i =1 λi 0, and Re λk 0, |Re λk | Im λk , k = 2, . . . , n. If Λ is realizable with prescribed elementary divisors
(λ − λ1 ), . . . , (λ − λ p )n p , (λ − λ p )n p , . . . , (λ − λk )nk ,
(6)
then for all t > 0 the list
= { λ p , Λ λ 1 , λ2 , . . . , λ p − 1 , λ p , λ p +2 , . . . , λn },
(7)
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λ1 = λ1 + 3t, λ p = Re λ p + i (Im λ p ± t ), is also realizable with elementary divisors where
(λ − λ1 ), . . . , (λ − λ p )n p , (λ − λ p )n p , . . . , (λ − λk )nk .
(8)
Proof. Let A be a nonnegative matrix with spectrum Λ and with the elementary divisors n in (6). Since Re λk 0 and |Re λk | Im λk , k = 2, . . . , n, then from Theorem 2.1, M = 0, m = − i =2 Re λi . Let A t in (7) and with the elementary divisors in (8). Without loss of be a matrix with the spectrum Λ generality we may take n p = 2 and it is enough to consider the following piece of A t
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
..
⎤ . Im λ p + t Re λ p
Re λ p −(Im λ p + t )
− Re λ p −(Im λ p + t )
⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦
Im λ p + t Re λ p
..
.
= t and m = m + 2t with and similarly for Im λ p − t. Then for all t > 0, we have from Theorem 2.1 M λ1 = λ1 + 3t. 2 | | Im λ p . Hence the new Perron eigenvalue is Remark 3.5. Observe that if we perturb k pairs of complex eigenvalues, the new Perron eigenvalue λ1 = λ1 + (k + 1)t. It is clear that Theorem 3.4 also holds for the lists in Corollary 3.1 and Theowill be rem 3.2 with the appropriate increments for λ1 . Note for example that for the list Λt+ in Corollary 3.1, if we perturb k pairs of complex eigenvalues the new Perron eigenvalue will be still λ1 + t. 4. Examples Example 4.1. We want to compute a nonnegative matrix A with spectrum
Λ = {16, −3, −3, −2 ± i , −2 ± i , −1 + 3i , −1 − 3i } and elementary divisors
(λ − 16), (λ + 3)2 , (λ + 2 − i )2 , (λ + 2 + i )2 , (λ + 1 ± 3i ). To do this, we consider the partition
Λ1 = {16, −1 + 3i , −1 − 3i }, Λ3 = {−3, −3},
Λ2 = {−2 ± i , −2 ± i },
Λ4 = φ
and the auxiliary lists
Γ2 = {8, −2 + i , −2 − i , −2 + i , −2 − i }, Γ3 = {6, −3, −3},
Γ4 = {0}.
Then we compute the matrix
⎡
8
B = ⎣ 45 8 0
0 6 16
8 35 8
⎤ ⎦
0
with spectrum Λ1 and diagonal entries 8, 6, 0 (see [13] about how to compute the matrix B). The matrices
R.L. Soto et al. / Linear Algebra and its Applications 439 (2013) 3591–3604
⎡
⎤
8 0 0 ⎢ 9 −2 1 ⎢ A 2 = ⎢ 12 −1 −2 ⎣9 0 0 11 0 0
0 0 0 0 ⎥ ⎥ −1 0 ⎥ , −2 1 ⎦ −1 −2
A3 =
6 0 10 −3 9 0
3603
0 −1 , −3
A 4 = [0],
have spectrum Γ2 , Γ3 , Γ4 , with the prescribed elementary divisors, respectively. Then
⎡
A1 A2
⎡ 0 ⎢ 1 ⎢ 4 ⎢ ⎢ 1 ⎢ ⎢ 3 =⎢ ⎢ 45 ⎢ 8 ⎢ 45 ⎢ ⎢ 8 ⎣ 45 8
0
1 ⎢1 ⎢ ⎢1 ⎢ ⎢1 ⎢ + ⎢1 ⎢ A3 ⎢0 ⎢ ⎢0 ⎣0 0
⎤
0 0 0 0⎥ ⎥ 0 0⎥⎡ ⎥ 0 0 ⎥ −8 2 2 2 2 0 0 0 ⎥ 45 0 0 0 0 −6 3 3 0 0⎥⎣ 8 ⎥ 1 0⎥ 0 0 0 0 0 16 0 0 ⎥ 1 0⎥ ⎦ 1 0 0 1
2 2 2 2 0 3 2 2 1 0 1 2 2 2 0 3 2 2 1 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 3 3
0
0
0
0
4
0 2
0 0
0 0
0 0
0 0
3 3 16 0
0 0
8 8 8 8 8
8 35 8
⎤ ⎦
0
⎤
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 35 ⎥ 8 ⎥ ⎥ 35 ⎥ 8 ⎥ 35 ⎦ 8
0
is nonnegative with spectrum Λ and the prescribed elementary divisors. Observe that in this example, the existence of a solution matrix also follows from Theorem 1.3 and the nonderogatory property of companion matrices. Example 4.2. Does there exist a nonnegative matrix with spectrum
Λ = {11, −1, −1 + 3i , −1 − 3i , −1 + 3i , −1 − 3i } and with elementary divisors
(λ − 11), (λ + 1), (λ + 1 − 3i )2 , (λ + 1 + 3i )2 ? To give an answer we observe that from Corollary 2.1 the list
{5, −1, −1 + i , −1 − i , −1 + i , −1 − i } is realizable with elementary divisors
(λ − 5), (λ + 1), (λ + 1 − i )2 , (λ + 1 + i )2 . Then from Theorem 3.4 and Remark 3.5, with t = 2, we have that Λ is realizable by a nonnegative matrix with the prescribed elementary divisors. Acknowledgement The authors would like to thank the referee for her/his helpful corrections and comments which greatly improved the presentation of the paper.
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References [1] A. Borobia, J. Moro, R.L. Soto, Negativity compensation in the nonnegative inverse eigenvalue problem, Linear Algebra Appl. 393 (2004) 73–89. [2] A. Brauer, Limits for the characteristic roots of a matrix IV: Applications to stochastic matrices, Duke Math. J. 19 (1952) 75–91. [3] W. Guo, Eigenvalues of nonnegative matrices, Linear Algebra Appl. 266 (1997) 261–270. [4] S. Guo, W. Guo, Perturbing non-real eigenvalues of nonnegative real matrices, Linear Algebra Appl. 426 (2007) 199–203. [5] T.J. Laffey, Perturbing non-real eigenvalues of nonnegative real matrices, Electron. J. Linear Algebra 12 (2005) 73–76. [6] T.J. Laffey, H. Šmigoc, Nonnegative realization of spectra having negative real parts, Linear Algebra Appl. 416 (2006) 148–159. [7] H. Minc, Inverse elementary divisor problem for nonnegative matrices, Proc. Amer. Math. Soc. 83 (4) (1981) 665–669. [8] H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988. [9] H. Perfect, Methods of constructing certain stochastic matrices II, Duke Math. J. 22 (1955) 305–311. [10] R.L. Soto, Existence and construction of nonnegative matrices with prescribed spectrum, Linear Algebra Appl. 369 (2003) 169–184. [11] R.L. Soto, O. Rojo, Applications of a Brauer Theorem in the nonnegative inverse eigenvalue problem, Linear Algebra Appl. 416 (2006) 844–856. [12] R.L. Soto, J. Ccapa, Nonnegative matrices with prescribed elementary divisors, Electron. J. Linear Algebra 17 (2008) 287–303. [13] R.L. Soto, O. Rojo, C.B. Manzaneda, On nonnegative realization of partitioned spectra, Electron. J. Linear Algebra 22 (2011) 557–572.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Nonnegative matrices with prescribed spectrum and elementary divisors ✩ Ricardo L. Soto a,∗ , Roberto C. Díaz a , Hans Nina a,b , Mario Salas a a b
Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile Departamento de Matemáticas, Universidad Mayor de San Andrés, La Paz, Bolivia
a r t i c l e
i n f o
Article history: Received 20 June 2013 Accepted 21 September 2013 Available online 16 October 2013 Submitted by R. Brualdi MSC: 15A29 15A18
a b s t r a c t In this paper we give new sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors, which drastically improve and contain some of the previous known conditions. We also show how to perturb complex eigenvalues of a nonnegative matrix while keeping its nonnegativity. These results allow us, under certain conditions, to easily decide if a given list is realizable with prescribed elementary divisors. © 2013 Elsevier Inc. All rights reserved.
Keywords: Nonnegative inverse elementary divisors problem Nonnegative matrices
1. Introduction Let A ∈ Cn×n and let
⎡J n1 (λ1 ) ⎢ J ( A ) = S −1 A S = ⎢ ⎣
⎤ ⎥ ⎥ ⎦
J n2 (λ2 )
..
. J nk (λk )
be the Jordan canonical form of A (hereafter JCF of A). The ni × ni submatrices ✩
*
Supported by Fondecyt 1120180, Chile, Mecesup UCN0711. Corresponding author. Fax: +56 55 355599. E-mail addresses: [email protected] (R.L. Soto), [email protected] (R.C. Díaz), [email protected] (H. Nina), [email protected] (M. Salas).
0024-3795/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.09.034
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⎡λ
i
⎢ ⎢ ⎣
λi
J ni (λi ) = ⎢
⎤
1
..
.
..
.
⎥ ⎥ ⎥, ⎦
1
i = 1, 2, . . . , k
λi are called the Jordan blocks of J ( A ). Then the elementary divisors of A are the polynomials (λ − λi )ni , that is, the characteristic polynomials of J ni (λi ), i = 1, . . . , k. The nonnegative inverse elementary divisors problem (NIEDP) is the problem of determining necessary and sufficient conditions under which the polynomials
(λ − λ1 )n1 , (λ − λ2 )n2 , . . . , (λ − λk )nk ,
n1 + · · · + nk = n,
are the elementary divisors of an n × n entrywise nonnegative matrix A [7,12]. The NIEDP contains the nonnegative inverse eigenvalue problem (NIEP), which asks for necessary and sufficient conditions for the existence of an entrywise nonnegative matrix with prescribed spectrum. Both problems remain unsolved. If Λ = {λ1 , λ2 , . . . , λn } is the spectrum of a nonnegative matrix A, we say that Λ is realizable and that A is the realizing matrix. A matrix A = (ai j )ni, j =1 is said to have constant row sums if all its rows sum up to the same constant, say
α , i.e.,
n
j =1 ai j
= α , i = 1, . . . , n. The set of all matrices with constant row sums equal
to α is denoted by CS α . It is clear that any matrix in CS α has the eigenvector e = (1, 1, . . . , 1) T corresponding to the eigenvalue α . Denote by ek the vector with one in the k-th position and zeros elsewhere. The relevance of matrices with constant row sums is due to the well-known fact that if Λ = {λ1 , λ2 , . . . , λn } is the spectrum of an n × n nonnegative matrix, then Λ is also the spectrum of a nonnegative matrix with constant row sums equal to its Perron eigenvalue. Let S be a nonsingular matrix such that S −1 A S = J ( A ) is the JCF of A. If A ∈ CS λ1 , then S can be chosen such that Se1 = e and the rows of S −1 = ( si , j ) satisfy n
s1, j = 1 and
n
s i , j = 0,
j =1
j =1
i = 2, . . . , n .
If T and S are n × n matrices of the form
⎡ ⎢
⎤ ··· ∗ . . .. ⎥ . .⎥ ∗ ⎥ .. .. ⎥ . . ∗⎦ .. . ∗ ∗ ∗
λ1
⎢0 T =⎢ ⎢ .. ⎣ . 0
⎡
1
s12
⎢1 S =⎢ ⎢ .. ⎣.
s22
1
sn2
⎢
and
with S being nonsingular, then S T S −1 e
..
.
⎤ · · · s1n .. . s2n ⎥ ⎥ ⎥ .. ⎥ , .. . . ⎦ .. . s nn
λ1 e, that is, S T S −1
= ∈ CS λ1 . We shall denote by σ ( A ) the spectrum of A, by E i j the n × n matrix with 1 in the (i , j ) position and zeros elsewhere, by rank( X ) the rank of the matrix X , and Im λ will be taken as nonnegative. Let E be the matrix E =±
E i , i +1 ,
K ⊂ {1, 2, . . . , n − 1}
i∈ K
and let A be an n × n complex matrix with JCF J ( A ) = S −1 A S. Then for an appropriate set K
J ( A ) + E = S −1 A S + E
= S −1 A + S E S −1 S
is the JCF of A + S E S −1 . Thus, given a complex matrix A with J ( A ) = S −1 A S, we may obtain, trivially, a matrix B = A + S E S −1 with σ ( B ) = σ ( A ) and prescribed elementary divisors. Moreover, if A ∈ CS λ1 and S = [e | ∗ | · · · | ∗], then ( A + S E S −1 ) ∈ CS λ1 .
R.L. Soto et al. / Linear Algebra and its Applications 439 (2013) 3591–3604
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The following two results, which we set here for completeness, have been shown to be very useful, not only to derive sufficient conditions for the realizability of both problems, the NIEP and the NIEDP, but for constructing a realizing matrix as well. The first result, due to Brauer [2], shows how to modify one single eigenvalue of a matrix, via a rank-one perturbation, without changing any of its remaining eigenvalues (see [10,1,12] and the references therein to see how Brauer’s result has been applied to NIEP and NIEDP). The second result, due to Rado and introduced by Perfect in [9] is an extension of Brauer’s result and shows how to change r eigenvalues of a matrix (r < n), via a perturbation of rank r, without changing any of its remaining n − r eigenvalues (see [9,11,13] to see how Rado’s result has been applied to NIEP). Theorem 1.1 (Brauer). (See [2].) Let A be an n × n arbitrary matrix with eigenvalues λ1 , λ2 , . . . , λn . Let v = ( v 1 , . . . , v n )T be an eigenvector of A corresponding to λk and let q be any n-dimensional vector. Then the matrix A + vq T has eigenvalues λ1 , . . . , λk−1 , λk + v T q, λk+1 , . . . , λn . Theorem 1.2 (Rado). (See [9].) Let A be an n × n arbitrary matrix with spectrum Λ = {λ1 , . . . , λn }. Let X = [x1 | · · · | xr ] be such that rank( X ) = r and Axi = λi xi , i = 1, . . . , r, r n. Let C be an r × n arbitrary matrix. Then A + X C has eigenvalues μ1 , . . . , μr , λr +1 , . . . , λn , where μ1 , . . . , μr are eigenvalues of the matrix Ω + C X with Ω = diag{λ1 , . . . , λr }. Next we recall the following results due to Laffey and Šmigoc [6], which will be used later: Theorem 1.3. (See [6].) Let Λ = {λ1 , λ2 , . . . , λn } be a list of complex numbers with Λ = Λ, Re λk 0, k = 2, . . . , n. Then Λ is the nonzero spectrum of a nonnegative matrix if
i)
s1 =
n
λk 0,
k =1
ii)
s2 =
n
λk2 > 0.
k =1
The minimal number N of zeros that need to be added to Λ to make it realizable satisfies s21 (n + N )s2 . Furthermore, the list Λ = {λ1 , . . . , λn , 0, . . . , 0} can be realized by a nonnegative matrix of the form C + α I , where C is a companion matrix with trace zero, α 0 and I is the identity matrix of the appropriate size. Theorem 1.4. (See [6].) Let Λ = {λ1 , λ2 , . . . , λn } be a list of complex numbers with Λ = Λ, Re λk 0, k = 2, . . . , n. Then Λ is realizable if and only if
i)
s1 =
n
λk 0,
k =1
ii)
s2 =
n
λk2 0,
k =1
iii)
s21 ns2 .
In [7] Minc proves that given an n × n diagonalizable positive matrix A, then there exists an n × n positive matrix B, with σ ( B ) = σ ( A ) and with arbitrarily prescribed elementary divisors, provided that elementary divisors corresponding to nonreal eigenvalues occur in conjugate pairs. As a consequence of this result and results from the NIEP for symmetric matrices, the authors in [12] give sufficient conditions for the existence and construction of an n × n positive matrix with prescribed spectrum and prescribed elementary divisors. The following result was given in [12]:
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Lemma 1.1. Let q be an arbitrary n-dimensional vector and let A ∈ CS λ1 with JCF J ( A ) = S −1 A S. Let λ1 + n q = λi , i = 2, . . . , n. Then A + eqT has JCF i i =1
J ( A) +
n
qi E 11 .
i =1
In particular, if
n
i =1 q i
= 0, then A and A + eqT are similar.
In [12] the authors completely solve the NIEDP in two particular cases: for lists of real numbers satisfying λ1 > λ2 · · · λn 0 and for Suleimanova lists λ1 > 0 λ2 · · · λn . For the general case, the authors give some sufficient conditions. The proofs are constructive, in the sense that if the conditions are satisfied then we can always construct a solution matrix. The paper is organized as follows: In Section 2, based on results of Brauer and Rado, we give new sufficient conditions for the NIEDP to have a solution, which drastically improve and contain known previous conditions in [12]. We prove, in Section 3, results which show how to perturb complex eigenvalues of a nonnegative matrix while keeping its nonnegativity. These results allow us, under certain conditions, to easily decide the realizability of a given list with prescribed elementary divisors. Finally, in Section 4, we show some examples to illustrate the results. 2. New conditions for the NIEDP We start this section with a general result, which gives a very simple sufficient condition for the existence and construction of a nonnegative matrix with prescribed spectrum and elementary divisors. This result contains Theorems 2, 3 and 4 in [12] as particular cases. Theorem 2.1. Let Λ = {λ1 , λ2 , . . . , λn } be a list of complex numbers with Λ = Λ, i = 2, . . . , n. Let
M = max {0, Re λi + Im λi }; 2i n
m=−
n
min{0, Re λi , Im λi }.
n
i =1 λi
0, and λ1 |λi |,
(1)
i =2
If
i ) λ1 M + m ,
(2)
when all possible Jordan blocks J ni (λi ), of size ni 2, are associated to a real eigenvalue λi < 0; or when there is at least one Jordan block J ni (λi ), of size ni 2, associated to a real eigenvalue λi 0 with M = Re λi 0 + Im λi 0 > |λk |, 2 k p, for some i 0 , p + 1 i 0 n − 1, or if
ii)
λ1 > M + m ,
(3)
when at least one Jordan block J ni (λi ), of size ni 2, is associated to a real eigenvalue λi 0 with M = λk 0, for some k, 2 k p, then there exists an n × n nonnegative matrix A ∈ CS λ1 with spectrum Λ and with prescribed elementary divisors
(λ − λ1 ), (λ − λ2 )n2 , . . . , (λ − λk )nk ,
n2 + · · · + nk = n − 1.
Proof. Let λ1 > λ2 · · · λ p be real numbers and let
λ p +1 , λ p +1 , . . . , λn−1 , λn−1 = λn be complex nonreal numbers. Let
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⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ J =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
3595
⎤
λ1 ..
. λp Re λ p +1
Im λ p +1
−Im λ p +1
Re λ p +1
..
. Re λn−1 −Im λn−1
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ Im λn−1 ⎦ Re λn−1
It is clear that the JCF of J is D = diag{λ1 , λ2 , . . . , λn }. Let E = i ∈ K E i ,i +1 , K ⊂ {2, . . . , n − 1} such that D + E is the prescribed JCF. Let S = [e | e2 | · · · | en ]. Then S E S −1 = i ∈ K (− E i ,1 + E i ,i +1 ) and B = S ( J + E ) S −1 is a matrix in CS λ1 with spectrum Λ and with the prescribed JCF. Now let q T = (q1 , q2 , . . . , qn ), where
q1 = −m, qk = −min{0, Re λk , −Im λk },
k = 2, 3, . . . , n .
We shall prove that A = B + eq is nonnegative. Let us first consider case i): If appears in at least one position (i , i + 1), i = 2, . . . , n − 1, then it is clear for < 0 with | | min{min2k p |λk |, min p +1kn Im λk }, that the entries of A in columns 2, 3, . . . , n, are all nonnegative. On the first column, the entries of A in positions k = 2, . . . , p are of the form λ1 − λk − m, or λ1 − λk − − m. Then they are all nonnegative if we choose < 0 with | | min2k p |λk |, whenever appears in at least one position (i , i + 1), i = 2, . . . , p. In positions k = p + 1, . . . , n, the entries on the first column of A are of the form λ1 − (Re λi + Im λi ) − m or λ1 − (Re λi − Im λi ) − m − , which from (2) are all nonnegative if we choose < 0, | | min p +1kn Im λk . If we have at least one Jordan block J ni (λi ), of size ni 2, associated to a real eigenvalue λi 0 and M occurs in a complex eigenvalue as M = Re λi 0 + Im λi 0 > |λk |, 2 k p, then we need that λ1 − λk − m. Hence we choose 0 < min2k p {λ1 − λk − m}. Then, in this case if we choose T
0 < min
min {λ1 − λk − m}, min |λk |,
2k p
2k p
min
p +1kn
Im λk ,
all entries in the first column of A are nonnegative. Now we consider case ii): If we choose
0 < min λ1 − ( M + m), min |λk |, 2k p
min
p +1kn
Im λk ,
the entries of A in columns 2, 3, . . . , n, are all nonnegative (for appearing in at least one position (i , i + 1)). In the first column, the entries of A in positions k = 2, . . . , p, are all nonnegative if we choose 0 < min{λ1 − ( M + m), min2k p |λk |}. In positions k = p + 1, . . . , n, if we choose 0 < min p +1kn Im λk the entries are all nonnegative. Then by choosing < 0 with | | min{min2k p |λk |, min p +1kn Im λk } or
0 < min
min {λ1 − λk − m}, min |λk |,
2k p
2k p
min
p +1kn
Im λk ,
in the case i) or by choosing
0 < min λ1 − ( M + m), min |λk |, 2k p
min
p +1kn
Im λk
in the case ii), all entries in the first column of A are nonnegative. It is clear from (2), that if does T not appear n in at least one position (i , i + 1), i = 2, . . . , n − 1, A = B + eq is still nonnegative. Finally, q = 0, from Lemma 1.1 A has the prescribed elementary divisors. 2 since i i =1
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The following corollary improves Theorem 4 in [12] and completely solves the NIEDP for complex lists Λ = {λ1 , λ2 , . . . , λn } with Re λi < 0, |Re λi | Im λi , i = 2, . . . , n. Corollary 2.1. Let Λ = {λ1 , λ2 , . . . , λn } be a list of complex numbers with Re λi < 0, |Re λi | Im λi , i = 2, . . . , n. Then there exists a nonnegative matrix A ∈ CS λ1 with spectrum Λ and with prescribed elementary divisors
(λ − λ1 ), (λ − λ2 )n2 , . . . , (λ − λk )nk ,
n2 + · · · + nk = n − 1
if and only if λ1 + λ2 + · · · + λn 0. Proof. The condition is necessary for the existence of a nonnegative matrix with spectrum Λ. From Theorem 2.1 the condition is sufficient. In fact,
M = max {0, Re λi + Im λi } = 0, 2i n
m=−
n
min{0, Re λi , Im λi } = −
i =2
n
Re λi = −
i =2
Then λ1 − m 0, that is, λ1 + λ2 + · · · + λn 0.
n
λi .
i =2
2
Remark 2.2. Remember that Rado’s Theorem 1.2 allows us to change r eigenvalues of a matrix A without changing any of its remaining n − r eigenvalues. When we apply Rado’s Theorem to obtain a nonnegative matrix A + X C with spectrum Λ = {λ1 , . . . , λn }, we start with a block diagonal matrix A where each block A k , k = 1, . . . , r, is nonnegative with Perron eigenvalue ωk and spectrum Γk = {ωk , λk2 , . . . , λkpk }. Then A + X C will be nonnegative with new eigenvalues λ1 , . . . , λr instead of ω1 , . . . , ωr . The r columns xi of X are linearly independent eigenvectors of A corresponding to the eigenvalues ωi and C is such that Ω + C X , with Ω = diag{ω1 , . . . , ωr }, has the new eigenvalues λ1 , . . . , λr . Thus, A + X C will have the desired spectrum Λ. The following result shows that the diagonal blocks A 1 , . . . , A r can have negative entries, which we may take away by setting appropriate entries in the suitable positions of the matrix C . Lemma 2.1. Let A be an n × n real block diagonal matrix, where each diagonal block A k ∈ CS λk1 (not necessarily nonnegative) has spectrum
Λk = {λk1 , λk2 , . . . , λkpk },
k = 1, 2, . . . , r ,
satisfying conditions of Theorem 2.1. Let qkT = (qk1 , . . . , qkpk ) with negative. Then M = A + X C is nonnegative with spectrum
pk
q j =1 kj
= 0, such that A k + e qkT is non-
{μ1 , μ2 , . . . , μr , λ12 , . . . , λ1p 1 , λ22 , . . . , λ2p 2 , . . . , λr2 , . . . , λrpr }, where μ1 , . . . , μr are eigenvalues of Ω + C X with Ω = diag{λ11 , λ21 , . . . , λr1 }. Proof. Let A = diag{ A 1 , A 2 , . . . , A r } and Axk = λk1 xk , k = 1, . . . , r, where xk has pk ones in positions p (k−1) + 1 to pk , with p 0 = 0, and zeros elsewhere. Let qk be an n-dimensional vector containing the qkT = (qk1 , qk2 , . . . , qkpk ) in positions p (k−1) + 1 to pk , with p 0 = 0 and zeros elsewhere. That is vector
xk = (0, . . . , 0, 1, . . . , 1, 0, . . . , 0) T ,
pk
qkT
= (0, . . . , 0, qk1 , . . . , qkpk , 0, . . . , 0). pk
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Let
X = [x1 | x2 | · · · | xr ]
and
C B = q1T q2T · · · qrT .
Then
M = A + X C B = x1 q1T + x2 q2T + · · · + xr qrT r is nonnegative with spectrum k=1 Λk . Observe that M is block diagonal with each diagonal block being nonnegative. Now, to change the eigenvalues λ11 , λ21 , . . . , λr1 to μ1 , μ2 , . . . , μr , we apply Rado’s Theorem to obtain a nonnegative matrix
M + X C R = ( A + X C B ) + X C R = A + X (C B + C R ) = A + X C , where C = C B + C R , with the prescribed spectrum.
2
Let A, X , C , and Ω be as in Theorem 1.2 and let S be nonsingular and partitioned as S = [ X | Y ] with S −1 = [ U ]. Then U X = I r , V Y = I n−r , V X = U Y = 0 and A X = X Ω . Hence we have V
S
−1
AS =
U V
S −1 X C S =
A[ X | Y ] = CX 0
and
S −1 ( A + X C ) S =
CY 0
U AY V AY
Ω 0
(4)
,
,
Ω +CX 0
C Y + U AY V AY
(5)
,
where B = Ω + C X is an r × r matrix with eigenvalues μ1 , . . . , μr (the new eigenvalues) and diagonal entries λ1 , . . . , λr (the former eigenvalues). Then from a result in [8, Chapter VI, Lemma 1.2], if Ω and V AY in (4) have no eigenvalues in common, A is similar to Ω ⊕ V AY . In the same way, if B = Ω + C X and V AY in (5) have no common eigenvalues, A + X C is similar to B ⊕ V AY . Then we have: Lemma 2.2. Let A, X , Y , V , C , and Ω be as above. If the matrices B = Ω + C X and V AY have no common eigenvalues, then
J ( A + X C ) = J ( B ) ⊕ J ( V AY ). In particular, if C X = 0, A and A + X C are similar. Now we extend Theorem 2.1: Theorem 2.3. Let Λ = {λ1 , λ2 , . . . , λn } be a list of complex numbers with Λ = Λ, λ1 maxi |λi |, i = 2, . . . , n; n λ 0. Let Λ = Λ1 ∪ · · · ∪ Λ p 1 +1 be a pairwise disjoint partition, with Λk = {λk1 , λk2 , . . . , λkpk }; i i =1 λ11 = λ1 , k = 1, . . . , p 1 + 1, where Λ1 is realizable, p 1 is the number of elements of the list Λ1 and some lists Λk can be empty. Let ω2 , . . . , ω p 1 +1 be real numbers satisfying 0 ωk λ1 , k = 2, . . . , p 1 + 1. Suppose that i) for each k = 2, . . . , p 1 + 1, there exists a list Γk = {ωk , λk1 , . . . , λkpk } with M k + mk , as in Theorem 2.1, where
M k = max {0, Re λki + Im λki }; 1 i p k
mk = −
pk
ωk Mk + mk or ωk >
min{0, Re λki , Im λki },
i =1
and ii) there exists a p 1 × p 1 nonnegative matrix B with spectrum Λ1 and diagonal entries ω2 , . . . , ω p 1 +1 .
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Then there exists an n × n nonnegative matrix M ∈ CS λ1 with spectrum Λ and with prescribed elementary divisors
(λ − λ1 ), (λ − λ2 )n2 , . . . , (λ − λk )nk , Proof. Let
⎡
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Jk = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
n2 + · · · + nk = n − 1.
⎤
ωk ..
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ −1 )
. λks Re λk(s+1) Im λk(s+1)
−Im λk(s+1) Re λk(s+1) ..
. Re λk( pk −1) Im λk( pk −1)
−Im λk( pk Re λk( pk −1)
with spectrum Γk and let E k = E i ,i +1 such that J k + E k is the prescribed JCF. Let S k = [e | e2 | · · · | e pk ]. Then A k = S k ( J k + E k ) S k−1 ∈ CS ωk , k = 2, . . . , p 1 + 1 and
⎡
⎤
A2
⎢
⎥ ⎥. ⎦
A3
A=⎢ ⎣
..
. A p 1 +1
Observe from Lemma 2.1 that for each k = 2, . . . , p 1 + 1, we have that A k + ek qkT is nonnegative for
p
k an appropriate vector qkT = (qk0 , qkp 1 , . . . , qkpk ) with q = 0 and from Lemma 1.1, A k + ek qkT j =0 kj has the prescribed elementary divisors. Then, we may apply Rado’s perturbation as in Lemma 2.1. Thus A + X C (with C = C B + C R ) is nonnegative with spectrum Λ and from Lemma 2.2 A has the prescribed elementary divisors. 2
Let us consider the following example: Example 2.1. Let Λ = {7, 1, −2, −2, −1 + 3i , −1 − 3i }. We construct a nonnegative matrix A with spectrum Λ and with elementary divisors
(λ − 7), (λ − 1), (λ + 2)2 , (λ + 1 − 3i ), (λ + 1 + 3i ). Consider the partition Λ = Λ1 ∪ Λ2 ∪ Λ3 ∪ Λ4 with
Λ1 = {7, −1 + 3i , −1 − 3i }; Γ2 = {4, −2, −2}; The matrix
⎡
4
B = ⎣ 34 7 0
0 1 7
3 8 7
Λ2 = {−2, −2};
Γ3 = {1, 1};
Λ3 = {1};
Λ4 = ∅ and
Γ4 = {0}.
⎤ ⎦
0
has the spectrum Λ1 with diagonal entries 4, 1, 0. Then from Theorem 2.3 we have
⎡
4 0 ⎢ 7 −2 ⎢ ⎢6 0
A=⎢
⎢ ⎣
0 −1 −2
⎤
⎡
1 ⎥ ⎢1 ⎥ ⎢ ⎥ ⎢1 ⎥+⎢ 1 0 ⎥ ⎢0 ⎦ ⎣0 0 1 0 0
⎤
0 0 0 0⎥⎡ ⎥ −4 0 0 ⎥ ⎣ 34 ⎥ 7 1 0⎥ ⎦ 0 1 0 0 1
2 0 0
2 0 0 0 0 0 0 7 0
3 8 7
0
⎤ ⎦
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⎡
0 ⎢ 3 ⎢ ⎢ 2
=⎢ ⎢ 34 ⎢ 7 ⎣ 34 7
0
2 2 0 1 2 0 0 0 0 0
0 0 0 1
0 0 0 0
0 0 1 0 7 0
3599
⎤
3 3⎥ ⎥ 3⎥ 8 7 8 7
⎥ ⎥ ⎥ ⎦
0
with the prescribed spectrum and elementary divisors. Corollary 2.2. Let Λ = {λ1 , . . . , λ p , λ p +1 , . . . , λs , λs+1 , . . . , λn } be a list of complex numbers with Λ = Λ, λ1 maxi |λi |, i = 2, . . . , n, ni=1 λi 0, where λ1 , . . . , λ p are real numbers, λ p +1 , . . . , λn are complex nonreal with Re λi 0, |Re λi | |Im λi |, for i = p + 1, . . . , s and Re λi 0 for i = s + 1, . . . , n. Let Λ = Λ1 ∪ Λ2 be a partition, where
Λ1 = {λ1 , λs+1 , . . . , λn }, Λ2 = {λ2 , . . . , λ p , λ p +1 , . . . , λs }, with Λ1 being realizable by a nonnegative companion matrix. Then Λ is realizable by a nonnegative matrix with any prescribed elementary divisors associated to the eigenvalues in Λ2 . Proof. From the hypothesis, there exists an (n − s + 1) × (n − s + 1) nonnegative companion matrix B with spectrum Λ1 and a diagonal element μ1 = b11 = tr B. It is clear from Corollary 2.1 that Γ2 = {μ1 , λ2 , . . . , λ p , λ p +1 , . . . , λs } is realizable by a nonnegative matrix A 2 with any prescribed elementary divisors. Let Γk = {0} for k = 3, . . . , n − s + 2. Then
⎡ ⎢
⎤
A2
A=⎢ ⎣
⎥ ⎥ ⎦
0
..
. 0
is nonnegative with spectrum
{μ1 , λ2 , . . . , λs , 0, . . . , 0 }. n−s
Now we apply Rado’s Theorem 1.2 to change μ1 , 0, . . . , 0 (n − s zeros) to λ1 , λs+1 , . . . , λn , and to obtain a nonnegative matrix M = A + X C with spectrum Λ and with the prescribed elementary divisors associated to the eigenvalues of Λ2 . 2 3. Complex eigenvalues perturbation Let Λ = {λ1 , λ2 , . . . , λn }, with λ2 ∈ R, be a list of complex numbers, which is realizable by a nonnegative matrix. Guo [3] proved that for all t > 0, the perturbed list Λt = {λ1 + t , λ2 ± t , λ3 , . . . , λn } is also realizable by a nonnegative matrix. The problem of perturbing the real parts of a pair of complex eigenvalues of a realizable list, by keeping the realizability of the new list, was considered by Laffey in [5] and by Guo and Guo in [4]. The following result shows that we can also perturb the imaginary parts of a pair of complex eigenvalues a + bi, a − bi of a realizable list, and to obtain a new list which is also realizable. Theorem 3.1. Let Λ = {λ1 , λ2 , . . . , λn } be a realizable list of complex numbers with Re λk 0, |Re λk | Im λk , k = 2, . . . , n. Let x p = Re λ p and y p = Im λ p , 2 p n − 1. Then for all t > 0, the perturbed lists
Λt+ = λ1 + 2t , λ2 , . . . , λ p −1 , x p + ( y p + t )i , x p − ( y p + t )i , λ p +2 , . . . , λn , Λt− = λ1 + 2t , λ2 , . . . , λ p −1 , x p + ( y p − t )i , x p − ( y p − t )i , λ p +2 , . . . , λn
are also realizable.
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Proof. Let λ1 , λ2 , . . . , λ p −1 be real numbers and let λ p , λ p , . . . , λn−1 , λn−1 be complex. Let us consider the matrix
⎡
⎤
λ1
⎢ ⎢ ⎢ ⎢ λ1 − λ p −1 ⎢ ⎢ λ1 − x p − y p − t Bt = ⎢ ⎢ λ1 − x p + y p + t ⎢ ⎢ ⎢ ⎢ ⎣λ − x −y 1
n −1
..
. λ p −1 xp
−( y p + t )
yp + t xp
..
n −1
λ1 − xn−1 + yn−1
Since Λ is realizable,
q1 =
n
n
i =1 λi
. xn−1
− y n −1
y n −1 xn−1
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
0. Let q T = (q1 , q2 , . . . , qn ), where
λi + t ,
i =1
qi = −λi ,
i = 2, . . . , p − 1,
q p = −x p + t , q j = −x j ,
q p +1 = − x p ,
j = p + 2, . . . , n .
Observe that B t ∈ CS λ1 , and since Re λk 0, |Re λk | Im λk , A = B t + eq T is nonnegative with spectrum
Λt+ = λ1 + 2t , λ2 , . . . , λ p −1 , x p + ( y p + t )i , x p − ( y p + t )i , λ p +2 , . . . , λn . It is clear that changing t to −t in the matrix B t and choosing q p = −x p and q p +1 = −x p + t, we obtain
Λt− = λ1 + 2t , λ2 , . . . , λ p −1 , x p + ( y p − t )i , x p − ( y p − t )i , λ p +2 , . . . , λn , which is also realizable.
2
Theorem 3.1 may be useful to decide the realizability of a list of complex numbers. Consider for example the list
Λ = {16, −2 + 5i , −2 − 5i , −3 + i , −3 − i }. It is clear from Corollary 2.1 (see also [1,12]) that the complex Suleimanova list
Λ = {6, −2 + 2i , −2 − 2i , −1 + i , −1 − i } is realizable. It is also realizable for example, by a nonnegative companion matrix. Then from Theorem 3.1 we have that the lists
Λ = {10, −2 + 2i , −2 − 2i , −3 + i , −3 − i } with t = 2, Λ = Λ = {16, −2 + 5i , −2 − 5i , −3 + i , −3 − i } with t = 3 are also realizable. We observe however, that with this procedure, the increase in the Perron eigenvalue may be greater than necessary. In fact, the Perron eigenvalue 16, in this example, may be replaced by λ1 10 since for λ1 = 10, the companion matrix with spectrum Λ is nonnegative. It is clear that we can also perturb the real parts of a pair of complex eigenvalues. In this case we have: Corollary 3.1. Let Λ = {λ1 , λ2 , . . . , λn } be a realizable list of complex numbers with Re λk 0, |Re λk | Im λk , k = 2, . . . , n. Let x p = Re λ p and y p = Im λ p , 2 p n − 1. Then for all t > 0, the perturbed lists
R.L. Soto et al. / Linear Algebra and its Applications 439 (2013) 3591–3604
3601
Λt+ = {λ1 + t , λ2 , . . . , λ p −1 , x p + t + iy p , x p + t − iy p , λ p +2 , . . . , λn } and
Λt− = {λ1 + 2t , λ2 , . . . , λ p −1 , x p − t + iy p , x p − t − iy p , λ p +2 , . . . , λn } are also realizable. For lists satisfying Theorem 1.4 we have: Theorem 3.2. Let Λ = {λ1 , λ2 , . . . , λn } be a list of complex numbers with Λ = Λ, Re λk 0, Im λk 0. If Λ is realizable, that is, if Λ satisfies conditions of Theorem 1.4, then for all t > 0,
Λt− = λ1 + 2t , λ2 , . . . , λ p −1 , x p + i ( y p − t ), x p − i ( y p − t ), λ p +2 , . . . , λn and
Λt+ = λ1 + 4t , λ2 , . . . , λ p −1 , x p + i ( y p + t ), x p − i ( y p + t ), λ p +2 , . . . , λn are also realizable (Λt− and Λt+ satisfy Theorem 1.4). Proof. Let x p = Re λ p and y p = Im λ p , 2 p n − 1. First we prove that Λt− is realizable. Denote by s1 (t ) the sum of the elements of the list Λt− and by s2 (t ) the sum of the squares of the elements of Λt− . Then
s1 (t ) = s1 + 2t , s2 (t ) = s2 + 4(λ1 + y p )t + 2t 2 . It is clear that Λt− = Λt− . Since s1 0, s2 0 and λ1 + y p 0, then s1 (t ) 0 and s2 (t ) 0. It remains to show that s21 (t ) ns2 (t ). Observe that
s21 (t ) = s21 + 4s1 t + 4t 2 , ns2 (t ) = ns2 + 4n(λ1 + y p )t + 2nt 2 . Then s21 ns2 , 4 < 4n, s1 (λ1 + y p ) (as s1 λ1 and y p 0) and 4 2n (as n 3). Hence s21 (t ) ns2 (t ). In the same way we may prove that Λt+ is realizable. 2 Remark 3.3. It is clear that in the previous theorems we can perturb simultaneously two or more pairs of conjugate complex numbers, under the condition that we appropriately increase λ1 as well. It is also clear that we may perturb a pair of real numbers λk , λk to obtain λk + ti, λk − ti. The results in this section give rise to an interesting, useful and easy procedure to construct, from a realizable list with certain elementary divisors, a new realizable list, where the elementary divisors associated to the new eigenvalues keep the structure of elementary divisors associated to the former eigenvalues. Theorem 3.4. Let Λ = {λ1 , λ2 , . . . , λn } be a list of complex numbers with Λ = Λ, λ1 maxi |λi |, i = 2, . . . , n, n i =1 λi 0, and Re λk 0, |Re λk | Im λk , k = 2, . . . , n. If Λ is realizable with prescribed elementary divisors
(λ − λ1 ), . . . , (λ − λ p )n p , (λ − λ p )n p , . . . , (λ − λk )nk ,
(6)
then for all t > 0 the list
= { λ p , Λ λ 1 , λ2 , . . . , λ p − 1 , λ p , λ p +2 , . . . , λn },
(7)
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R.L. Soto et al. / Linear Algebra and its Applications 439 (2013) 3591–3604
λ1 = λ1 + 3t, λ p = Re λ p + i (Im λ p ± t ), is also realizable with elementary divisors where
(λ − λ1 ), . . . , (λ − λ p )n p , (λ − λ p )n p , . . . , (λ − λk )nk .
(8)
Proof. Let A be a nonnegative matrix with spectrum Λ and with the elementary divisors n in (6). Since Re λk 0 and |Re λk | Im λk , k = 2, . . . , n, then from Theorem 2.1, M = 0, m = − i =2 Re λi . Let A t in (7) and with the elementary divisors in (8). Without loss of be a matrix with the spectrum Λ generality we may take n p = 2 and it is enough to consider the following piece of A t
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
..
⎤ . Im λ p + t Re λ p
Re λ p −(Im λ p + t )
− Re λ p −(Im λ p + t )
⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦
Im λ p + t Re λ p
..
.
= t and m = m + 2t with and similarly for Im λ p − t. Then for all t > 0, we have from Theorem 2.1 M λ1 = λ1 + 3t. 2 | | Im λ p . Hence the new Perron eigenvalue is Remark 3.5. Observe that if we perturb k pairs of complex eigenvalues, the new Perron eigenvalue λ1 = λ1 + (k + 1)t. It is clear that Theorem 3.4 also holds for the lists in Corollary 3.1 and Theowill be rem 3.2 with the appropriate increments for λ1 . Note for example that for the list Λt+ in Corollary 3.1, if we perturb k pairs of complex eigenvalues the new Perron eigenvalue will be still λ1 + t. 4. Examples Example 4.1. We want to compute a nonnegative matrix A with spectrum
Λ = {16, −3, −3, −2 ± i , −2 ± i , −1 + 3i , −1 − 3i } and elementary divisors
(λ − 16), (λ + 3)2 , (λ + 2 − i )2 , (λ + 2 + i )2 , (λ + 1 ± 3i ). To do this, we consider the partition
Λ1 = {16, −1 + 3i , −1 − 3i }, Λ3 = {−3, −3},
Λ2 = {−2 ± i , −2 ± i },
Λ4 = φ
and the auxiliary lists
Γ2 = {8, −2 + i , −2 − i , −2 + i , −2 − i }, Γ3 = {6, −3, −3},
Γ4 = {0}.
Then we compute the matrix
⎡
8
B = ⎣ 45 8 0
0 6 16
8 35 8
⎤ ⎦
0
with spectrum Λ1 and diagonal entries 8, 6, 0 (see [13] about how to compute the matrix B). The matrices
R.L. Soto et al. / Linear Algebra and its Applications 439 (2013) 3591–3604
⎡
⎤
8 0 0 ⎢ 9 −2 1 ⎢ A 2 = ⎢ 12 −1 −2 ⎣9 0 0 11 0 0
0 0 0 0 ⎥ ⎥ −1 0 ⎥ , −2 1 ⎦ −1 −2
A3 =
6 0 10 −3 9 0
3603
0 −1 , −3
A 4 = [0],
have spectrum Γ2 , Γ3 , Γ4 , with the prescribed elementary divisors, respectively. Then
⎡
A1 A2
⎡ 0 ⎢ 1 ⎢ 4 ⎢ ⎢ 1 ⎢ ⎢ 3 =⎢ ⎢ 45 ⎢ 8 ⎢ 45 ⎢ ⎢ 8 ⎣ 45 8
0
1 ⎢1 ⎢ ⎢1 ⎢ ⎢1 ⎢ + ⎢1 ⎢ A3 ⎢0 ⎢ ⎢0 ⎣0 0
⎤
0 0 0 0⎥ ⎥ 0 0⎥⎡ ⎥ 0 0 ⎥ −8 2 2 2 2 0 0 0 ⎥ 45 0 0 0 0 −6 3 3 0 0⎥⎣ 8 ⎥ 1 0⎥ 0 0 0 0 0 16 0 0 ⎥ 1 0⎥ ⎦ 1 0 0 1
2 2 2 2 0 3 2 2 1 0 1 2 2 2 0 3 2 2 1 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 3 3
0
0
0
0
4
0 2
0 0
0 0
0 0
0 0
3 3 16 0
0 0
8 8 8 8 8
8 35 8
⎤ ⎦
0
⎤
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 35 ⎥ 8 ⎥ ⎥ 35 ⎥ 8 ⎥ 35 ⎦ 8
0
is nonnegative with spectrum Λ and the prescribed elementary divisors. Observe that in this example, the existence of a solution matrix also follows from Theorem 1.3 and the nonderogatory property of companion matrices. Example 4.2. Does there exist a nonnegative matrix with spectrum
Λ = {11, −1, −1 + 3i , −1 − 3i , −1 + 3i , −1 − 3i } and with elementary divisors
(λ − 11), (λ + 1), (λ + 1 − 3i )2 , (λ + 1 + 3i )2 ? To give an answer we observe that from Corollary 2.1 the list
{5, −1, −1 + i , −1 − i , −1 + i , −1 − i } is realizable with elementary divisors
(λ − 5), (λ + 1), (λ + 1 − i )2 , (λ + 1 + i )2 . Then from Theorem 3.4 and Remark 3.5, with t = 2, we have that Λ is realizable by a nonnegative matrix with the prescribed elementary divisors. Acknowledgement The authors would like to thank the referee for her/his helpful corrections and comments which greatly improved the presentation of the paper.
3604
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