Spec. Matrices 2018; 6:301–309
Research Article Open Access
Macarena Collao, Mario Salas, and Ricardo L. Soto* Spectra universally realizable by doubly stochastic matrices https://doi.org/10.1515/spma-2018-0025 Received November 27, 2017; accepted May 26, 2018
Abstract: A list of complex numbers Λ = {λ1, ... , λn} is said to be realizable if it is the spectrum of an en- trywise nonnegative matrix, and universally realizable if there exists a nonnegative matrix with spectrum Λ for each Jordan canonical form associated with Λ. The problem of characterizing the lists which are univer- sally realizable is called the nonnegative inverse elementary divisors problem (NIEDP). This is a hard problem, which remains unsolved. A complete solution, if any, is still far from the current state of the art in the prob- lem. In particular, in this paper we consider the NIEDP for generalized doubly stochastic matrices, and give new su cient conditions for the existence and construction of a solution matrix. These conditions improve those given in [ELA 30 (2015) 704-720].
Keywords: Elementary divisor, nonnegative matrix, nonnegative inverse elementary divisors problem
MSC: 15A18, 15A51
1 Introduction
k A ∈ n×n J A S−1AS ⊕ J Jordan canonical form A JCF A Let C and let ( ) = = ni(λi) be a of (hereafter, the of ). The i=1 ni × ni submatrices λi 1 . λ .. J λ i i ... k ni ( i) = , = 1, 2, , , .. . 1 λi
ni are the Jordan blocks of J(A). The elementary divisors of A are the polynomials (λ − λi) , that is, the charac- teristic polynomials of Jni (λi), i = 1, ... , k.
The nonnegative inverse elementary divisors problem (hereafter, the NIEDP) is the problem of determining nec- essary and su cient conditions under which there exists a nonnegative matrix with prescribed elementary divisors (see [5, 8, 9, 12, 14–16]). The NIEDP is closely related to the nonnegative inverse eigenvalue problem (hereafter, the NIEP), which is the problem of determining necessary and su cient conditions for a list of complex numbers Λ = {λ1, λ2, ... , λn} to be the spectrum of an n × n entrywise nonnegative matrix. If there exists a nonnegative matrix A with spectrum Λ, we say that Λ is realizable and that A is the realizing matrix. Following de nition and notation in [4], we shall say that Λ is universally realizable (UR) if there exists a non-
Macarena Collao: Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile, E-mail: [email protected] Mario Salas: Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile, E-mail: [email protected] *Corresponding Author: Ricardo L. Soto: Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile, E-mail: [email protected]
Open Access. © 2018 Macarena Collao et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution- NonCommercial-NoDerivs 4.0 License. 302 Ë Macarena Collao et al. negative matrix with spectrum Λ for each possible JCF associated with Λ. Both problems, the NIEDP and the NIEP, remain unsolved.
In [12] it has been proved that lists of real numbers Λ = {λ1, ... , λn} with
i) λ1 > λ2 ≥ · · · ≥ λn ≥ 0 (1) and ii) λ1 > 0 > λ2 ≥ · · · ≥ λn (2)
are UR. In [5, 14] it has been proved, respectively, that lists of complex numbers Λ = {λ1, ... , λn} with
iii) λ1 > 0, Re λi < 0, |Re λi| ≥ |Im λi| , i = 2, ... , n, (3) and √
iv) λ1 > 0, Re λi < 0, 3 Re λi ≥ |Im λi| , i = 2, ... , n (4) n X are also UR. In (2), (3) and (4), the necessary and su cient condition is λi ≥ 0. In [4] the authors prove that i=1 real spectra with two positive eigenvalues of the form {p, q, −r, −r, ... , −r}, with p, q, r > 0, p+q+(n−2)r = 0, p > q, r, and q ≤ r, are also UR. In all these cases the spectra are diagonalizably realizable (DR). Thus, it is natural to ask if all DR lists are also UR. This is not true in the case of doubly stochastic matrices. In fact, Minc [9] has showed that the doubly stochastic 0 1 1 1 1 0 1 2 1 1 0
1 1 λ 1 2 has eigenvalues 1, − 2 , − 2 , but no 3×3 doubly stochastic matrix has the elementary divisor ( + 2 ) . For more general lists, su cient conditions have been obtained in [3, 8, 9, 12, 14].
T The set of all matrices with constant row sums equal to γ is denoted by CSγ . It is clear that e = [1, 1, ... , 1] is an eigenvector of any matrix A ∈ CSγ , corresponding to the eigenvalue γ. The relevance of the real matri- ces with constant row sums is due to the well known fact that the problem of nding a nonnegative matrix with spectrum Λ = {λ1, ... , λn}, λ1 being the Perron eigenvalue, is equivalent to the problem of nding a CS Λ nonnegative matrix in λ1 with spectrum .
We mention the following perturbation result due to Brauer [2, Theorem 27], which shows how to change a single eigenvalue of an n × n matrix, via a rank-one perturbation, without changing any of the remaining n − 1 eigenvalues. This result has been employed, in connection with the NIEP and the NIEDP, to derive suf- cient conditions for the existence and construction of nonnegative matrices with prescribed spectrum and prescribed elementary divisors (see [3, 10–14] and the references therein).
Theorem 1.1. [2] Let A be an n × n arbitrary matrix with eigenvalues T λ1, λ2, ... , λn . Let v = [v1, v2, ... , vn] be an eigenvector of A associated with the eigenvalue λk and let q be T T any n-dimensional vector. Then the matrix A + vq has eigenvalues λ1, λ2, ... , λk−1, λk + v q, λk+1, ... , λn .
We shall denote by Ei,j the n × n matrix with 1 in the (i, j) position and zeros elsewhere. The following lemma in [12], shows the JCF of Brauer’s perturbation A + eqT .
Lemma 1.1. [12] Let A ∈ CS be with JCF λ1
−1 J(A) = S AS = diag{J1(λ1), Jn2 (λ2), ... , Jnk (λk)} Spectra universally realizable by doubly stochastic matrices Ë 303
n n T P T P Let q = [q1, ... , qn] and λ1 + qi ≠ λi , i = 2, ... , n. Then A +eq has JCFJ(A)+ qi E11. In particular, i=1 i=1 n P T if qi = 0, then A and A + eq are similar. i=1
In [8, Theorem 1], Minc proved the following result:
Theorem 1.2. [8] Let Λ = {λ1, λ2, ... , λn} be a list of complex numbers, which is realizable by a diagonalizable positive (diagonalizable positive doubly stochastic) matrix A. Then, for each JCFJΛ associated with Λ, there exists a positive (positive doubly stochastic) matrix B, with the same spectrum as A, and with JCFJ(B) = JΛ .
The problem of nding a nonnegative matrix with spectrum Λ = {λ1, ... , λn} for each possible JCF associ- ated with Λ (NIEDP) is a very hard problem. As with the NIEP, it remains unsolved. A complete solution, if any, is still far from the current state of the art about the problem. For that reason, we identi ed some open topics to work on, for which we have a reasonable chance of success. In particular, the NIEDP for generalized doubly stochastic matrices. In Section 2, we discuss certain results relating nonnegative with positive matri- ces. In particular, on nonnegative matrices which are cospectral (similar) to positive matrices. In Section 3 we consider the NIEDP for generalized doubly stochastic matrices and give new su cient conditions for the existence of a solution. These conditions improve the su cient conditions in [16]. Moreover, we show that, under certain restrictions, Theorem 1.2 of Minc holds for nonnegative matrices and that the diagonalizablity condition can be removed.
2 On nonnegative matrices similar to positive matrices
In [1, 7] the authors prove that if A is a nonnegative irreducible matrix with a positive row or column, then A is similar to a positive matrix. Next, we show, in a simple way, that this similarity preserves the positive A ∈ CS A ∈ CS row or column and that the irreducibility condition is not necessary if λ1 . Moreover, if λ1 is diagonalizable, it is similar to a diagonalizable positive matrix. As a consequence, if Λ = {λ1, ... , λn} A ∈ CS Λ is the spectrum of a nonnegative diagonalizable matrix λ1 with a positive row or column, then is universally realizable. First we consider the following result, which is often attributed to Johnson [6].
Lemma 2.1. Let A be a nonnegative matrix with spectrum Λ = {λ1, ... , λn}, λ1 being the Perron eigenvalue. Then A is cospectral to a nonnegative matrix B ∈ CS λ1 .
Theorem 2.1. Let A be a nonnegative matrix with spectrum Λ = {λ1, ... , λn}, λ1 > |λi| , i = 2, ... , n, and a positive row or column. If A is irreducible (respectively reducible), then A is similar (respectively cospectral) to a nonnegative matrix B ∈ CS with a positive row or column. λ1 Proof. If A = ai,j is irreducible nonnegative with a positive row or column, it has a positive Perron eigenvec- h i T −1 xj x x ... xn D diag{x ... xn} B D AD a ∈ CS tor = [ 1, , ] . Let = 1, , . Then = = xi i,j λ1 is nonnegative, similar to A, and with a positive row or column. If A is an arbitrary nonnegative matrix with a positive column, then our proof reproduces Johnson’s proof of Lemma 2.1 for the case λ1 > |λi| , i = 2, ... , n, and a positive column. It is clear that the nonnegative matrix B ∈ CS A AT λ1 has a positive column. If is nonnegative with a positive row, we take and the result follows.
Corollary 2.1. Let A be an nonnegative matrix with spectrum
Λ = {λ1, ... , λn}, λ1 > |λi| , i = 2, ... , n, and a positive row or column. Then A is cospectral to a positive matrix. If A ∈ CS then A is similar to a positive matrix. Moreover, if A is diagonalizable, then it is similar to λ1 , a diagonalizable positive matrix. 304 Ë Macarena Collao et al.
Proof. Let A be nonnegative with a positive column. Then, from Theorem 2.1, A is cospectral to a nonnegative B ∈ CS b b ... b T matrix λ1 with a positive column, say the last column [ 1n , 2n , , nn] . Let
qT q ... q q i ... n = [ 1, , n] , i > 0, = 1, , − 1, n n P P−1 (5) qi = 0, qi < min bin . i n i=1 i=1 1≤ ≤
Then M = B + eqT is positive and from Lemma 1.1 it is similar to B and cospectral to A. If A is nonnegative with a positive row, then we take AT , which is nonnegative with a positive column. Then AT is cospectral to B ∈ CS A ∈ CS a nonnegative matrix λ1 with a positive column, and the result follows. If the last column of λ1 is positive, then M = A + eqT , where qT is as in (5), is positive and from Lemma 1.1, it is similar to A. If A has a positive row, we take AT and the result follows as before. If A is diagonalizable, then from Lemma 1.1 M is also diagonalizable.
As an example, we have that the reducible nonnegative matrix 0 1 0 0 3 1 0 0 0 3 A = 0 0 2 1 1 ∈ CS4 0 0 1 2 1 0 0 1 1 2
M A eqT qT 1 1 1 1 4 is similar to the positive matrix = + , where = 5 , 5 , 5 , 5 , − 5 .
Corollary 2.2. Let A (A ∈ CS ) be a nonnegative matrix with spectrum Λ {λ λ ... λ } λ |λ | λ1 = 1, 2, , n , 1 > i , i = 2, ... , n, and having fewer than n zero entries. Then A is cospectral (similar) to a positive matrix.
Proof. It is clear that if A has fewer than n zero entries, then A has a positive row or column. Thus, from A A ∈ CS Corollary 2.1, is cospectral to a positive matrix, and if λ1 , then it is similar to a positive matrix.
In [7] the authors de ne a Perron extreme spectrum as follows:
De nition 2.1. A realizable list of complex numbers Λ = {λ1, λ2, ... , λn}, with the Perron eigenvalue λ1, is Perron extreme if the list Λ− = {λ1 − , λ2, ... , λn} is not realizable for every > 0.
We shall say that a matrix is Perron extreme if its spectrum is Perron extreme. All nonnegative matrices with trace zero are Perron extreme.
Lemma 2.2. Let Λ = {λ1, λ2, ... , λn} be a realizable list of complex numbers. Then Λ is not Perron extreme if and only if Λ is the spectrum of a positive matrix.
Proof. If Λ is not Perron extreme, then there exists 0 > 0 such that {λ1 − , λ2, ... , λn}, with 0 < ≤ 0, is B B0 ∈ CS realizable by a nonnegative matrix , which from Lemma 2.1, is cospectral to a matrix λ1−. Then for T 0 T q = n , n , ... , n , A = B + eq is positive with spectrum Λ. Λ A A A0 a0 ∈ CS If is the spectrum of a positive matrix , then is similar to a positive matrix = ij λ1 . Then T 0 0 T for q = [−, 0, ... , 0] , with 0 < ≤ minai1, we have that A + eq is nonnegative with spectrum Λ− = 1≤i≤n {λ1 − , λ2, ... , λn}. Thus, Λ is not Perron extreme. Spectra universally realizable by doubly stochastic matrices Ë 305
3 Lists universally realizable by nonnegative generalized doubly stochastic matrices
In [3] the authors introduce the n × n nonsingular matrix 1 1 1 ··· 1 1 1 1 1 ··· 1 −1 1 1 1 · · · −2 0 S = (6) . . . . . . ······ . . 1 1 −(n − 2) ··· 0 0 1 −(n − 1) 0 ··· 0 0 with rows
R1 = (1, 1, ... , 1)
Rk = 1, 1, ... , 1, −(k − 1), 0, ... , 0 , k = 2, ... , n, | {z } | {z } n−(k−1) ones k−2 zeros and S−1 with columns T 1 1 1 1 1 C = , , , ... , , 1 n n(n − 1) (n − 1)(n − 2) 3 · 2 2 · 1 T 1 1 1 1 1 Ck = , , , ... , , − , 0, ... , 0 , k = 2, ... , n. n n(n − 1) (n − 1)(n − 2) (k + 1)k k | {z } k−2 zeros
In terms of a real list Λ = {λ1, λ2, ... , λn}, the authors in [3] prove that if n − r + 2 − Tr−1 < λr < (n − r + 2) Tr−1, r = 2, . . . n, (7) n − r + 1 where r−1 λ1 X λp Tr−1 = + , (8) n (n − p + 2)(n − p + 1) p=2
λ1 −1 r = 3, ... , n, with T1 = n , and D = diag {λ1, λ2, ... , λn} , then B = SDS is a positive symmetric gen- eralized doubly stochastic matrix with spectrum Λ. In [3] it is also shown that if D = diag{λ1, ... , λn} with −1 λ1 > λ2 ≥ · · · ≥ λn ≥ 0, then λr , r = 2, . . . n, satis es (7) and SDS is a positive symmetric generalized doubly stochastic matrix. Thus, from Theorem 1.2, Λ is UR. Then, we have:
Theorem 3.1. Let Λ = {λ1, λ2, ... , λn} be a list of real numbers with
λ1 > λ2 ≥ · · · ≥ λn ≥ 0. Then Λ is UR by positive generalized doubly stochastic matrices.
Now, we show that if Λ = {λ1, λ2, ... , λn} is a list of real numbers of "Suleimanova" type, then under certain conditions, Λ is the spectrum of a positive diagonalizable generalized doubly stochastic matrix, and as a consequence Λ is UR by positive generalized doubly stochastic matrices.
Theorem 3.2. Let Λ = {λ1, λ2, ... , λn} be a list of real numbers with λ1 > 0 > λ2 ≥ · · · ≥ λn and let D = diag {λ1, λ2, ... , λn} . If r−1 X λp n (n − r + 1) λ1 > −n − λr (9) (n − p + 2)(n − p + 1) n − r + 2 p=2 306 Ë Macarena Collao et al. for r = 2, 3, ... , n, where the sum on the right side is zero for r = 2, then SDS−1 is a positive symmetric generalized doubly stochastic matrix with spectrum Λ.
Proof. Observe that inequality (9) is the rst inequality in (7), in terms of λ1. Then n − r + 2 − Tr < λr . n − r + 1 −1
Since nλr < 0 and n − r + 2 > 0, then
n n (n − r + 1) λr < − λr . n − r + 2 n − r + 2 Therefore, from (9) we have
r−1 X λp n −n + λr < λ1 (n − p + 2)(n − p + 1) n − r + 2 p=2 and then λr < (n − r + 2) Tr−1.
Thus, λr satis es (7), r = 2, ... , n, and the result follows.
The following corollary simpli es the condition (9):
Corollary 3.1. Let Λ = {λ1, λ2, ... , λn} be a list of real numbers with λ1 > 0 > λ2 ≥ · · · ≥ λn and let D = diag {λ1, λ2, ... , λn} . If n λ n P λp 1 > − (n−p+1)(n−p+2) , (10) p=2 then SDS−1 is a positive symmetric generalized doubly stochastic matrix with spectrum Λ.
Proof. Let r−1 X λp n (n − r + 1) f(r) = −n − λr , (n − p + 2)(n − p + 1) n − r + 2 p=2 r = 2, ... , n, with the rst term on the right side being zero for r = 2. We shall prove that if r1 < r2, then f(r1) < f(r2). In fact,
f (r) − f (r − 1) = r−2 r−1 P λp P λp (n−p+1)(n−p+2) − (n−p+1)(n−p+2) = n p=2 p=2 λ n−r+2 λ n−r+1 + r−1 n−r+3 − r n−r+2 n − r + 1 = −n (λr − λr−1) > 0. n − r + 2 Thus, f(n) > f (n − 1) > ··· > f(3) > f(2), and the result follows.
The following lemma shows that, for a list of real numbers of "Suleimanova" type, Corollary 3.1 gives a better result than Corollary 2.5 in [16]:
Lemma 3.1. The lower bound (10) of Corollary 3.1 is less than or equal to the lower bound of Corollary 3.5 in [16]. Spectra universally realizable by doubly stochastic matrices Ë 307
Proof.
n n λ n P p n P λn − (n−p+1)(n−p+2) ≤ − (n−p+1)(n−p+2) p=2 p=2 = −(n − 1)(λn).
There are many examples which show that Theorem 1.2 holds for diagonalizable nonnegative (diagonalizable nonnegative generalized doubly stochastic) matrices. For instance, the lists in (1), (2), (3), and (4) are all UR. Moreover, if Λ = {λ1, ... , λn} is a list of complex numbers, which is realizable by a diagonalizable nonnega- tive matrix A satisfying the hypothesis of Corollaries 2.1 or 2.2, then Λ is UR. In the same way, the realizable lists of Theorem 3.1 and Theorem 3.2 are also UR. However, we do not know if Theorem 1.2 holds for a gen- eral diagonalizable nonnegative (diagonalizable nonnegative generalized doubly stochastic) matrix. Here we introduce other results on universal realizability and discuss the positivity and diagonalizability conditions of Theorem 1.2, and we nd, under certain restrictions, some extensions of the Minc result to nonnegative matrices. We start with the following result:
Theorem 3.3. Let Λ = {λ1, ... , λn}, λ1 > |λi| , i = 2, ... , n, be a list of complex numbers which is the spectrum of an n n diagonalizable nonnegative matrix A ∈ CS with a positive row or column. Then Λ is UR. × λ1
Proof. From Corollary 2.1, A is similar to a diagonalizable positive matrix Ab. Then, from Theorem 1.2, there exists an n × n nonnegative matrix B with the same spectrum as A for each JCF associated with Λ.
Observe that if A is nonnegative with spectrum Λ = {λ1, ... , λn}, then we may assume that A has at least one zero entry in each column, otherwise A has a positive column. The following result generates a sequence of nonnegative matrices with the desired JCF, and with the Perron eigenvalue being λ1 + for all > 0.
Theorem 3.4. Let Λ = {λ1, ... , λn} be a list of complex numbers which is the spectrum of an n × n diagonal- izable nonnegative generalized doubly stochastic matrix A. Then, for all > 0, Λ = {λ1 + , λ2, ... , λn} is UR.
Proof. A ∈ CS Let > 0. since λ1 , there exists a positive vector qT q ... q q i ... n = ( 1, , n) with i = n , = 1, , , such that B = A + eqT is a positive generalized doubly stochastic matrix with spectrum
Λ {λ λ ... λ } B ∈ CS = 1 + , 2, , n and λ1+.
From Lemma 1.1, B is diagonalizable and from Theorem 1.2, Λ is UR.
Example 3.1. Consider the matrix 0 2 2 1 1 1 2 2 0 2 1 1 1 2 2 2 0 1 1 1 2 A = 2 2 2 0 0 1 2 ∈ CS , 9 0 2 2 2 0 1 2 2 2 2 1 1 0 1 0 2 2 1 1 2 1 which is diagonalizable nonnegative with a positive column and spectrum Λ = {9, −2, −2, −1 ± i, −1 ± i}. Then, from Theorem 3.3, Λ is UR. 308 Ë Macarena Collao et al.
Remark 3.1. Observe that if we want that a realizable list Λ = {λ1, ... , λn} to be the spectrum of a nonnega- tive matrix A ∈ CS with a positive column, then the Perron eigenvalue λ must be simple. In fact, if λ has λ1 , 1 1 algebraic multiplicity ma ≥ 2, for instance, then Λ has a reducible nonnegative realization A, and therefore A cannot have a positive column.
Now, we show that under certain restrictions, the requirement that A is a diagonalizable matrix can be re- moved from the statement of Theorem 1.2. However, our results do not allow us to realize, universally, the list Λ.
Theorem 3.5. Let A be an n × n nonnegative matrix with spectrum Λ = {λ1, ... , λn} and a positive row or column. Let J(A) be a JCF of A. Then there exists a nonnegative matrix C with spectrum Λ for each possible modi cation of J(A), which does not include elementary divisors of smaller size than the size of the elementary divisors A.
Proof. A a ∈ CS We assume that = i,j λ1 with its last column being positive and that
J(A) = diag{Jn1 (λ1), Jn2 (λ2), ... , Jnk (λk)}, n1 + ··· + nk = n, qT q q ... q B A eqT ∈ CS is not a diagonal matrix. Let = [ 1, 2, , n] be as in (5). Then = ( + ) λ1 is positive, −1 P and from Lemma 1.1 J(B) = J(A). Let S be a nonsingular matrix such that S BS = J(B). Let E = Ei,i+1, i∈K K ⊂ {2, ... , n−1} be such that J(A)+E is the desired JCF. Let δ > 0 be small enough such that B+δSES−1 ≥ 0. Then the matrix C = B + δSES−1 = SJ(B)S−1 + δSES−1 = S(J(B) + δE)S−1 −1 is nonnegative with spectrum Λ, and with the desired elementary divisors. Besides, since SES ∈ CS0, C ∈ CS λ1 .
Corollary 3.2. Let A ∈ CS be an n n nonnegative matrix with spectrum Λ {λ ... λ } and JCFJ A λ1 × = 1, , n ( ). Then for all > 0 there exists a nonnegative matrix C with spectrum Λ = {λ1 + , ... , λn}, for each possible modi cation of J(A), which does not include elementary divisors of smaller size than the size of the elementary divisors A.
Example 3.2. The matrix 0 5 0 0 1 6 0 0 0 0 5 0 0 0 1 0 3 1 0 0 A = 1 0 0 5 0 has JCFJ(A) = 0 0 3 0 0 . 1 0 5 0 0 0 0 0 −5 0 0 0 4 0 2 0 0 0 0 −5 Let = 0.001. We shall use Corollary 3.2 to construct a nonnegative matrix C with JCFJ(C) = diag{J1(6.001), J2(3), J2(−5)}. First we compute a positive matrix B ∈ CS6.001 : 1 25 001 1 1 5001 5000 5000 5000 5000 5000 25 001 1 1 1 5001 5000 5000 5000 5000 5000 0.001 T B = A + ee = 5001 1 1 25 001 1 . 5 5000 5000 5000 5000 5000 5001 1 25 001 1 1 5000 5000 5000 5000 5000 1 1 20 001 1 10 001 5000 5000 5000 5000 5000 −1 −1 Next, we compute the matrix S such that S BS = J(B). Then, for E4,5 and δ small enough, C = B+δSE4,5S will be nonnegative in CS6.001 with the desired JCF. Obviously the matrix S is necessary to choose an appropriate δ, which in this example is δ = 0.000001. Spectra universally realizable by doubly stochastic matrices Ë 309
Acknowledgement: Work supported by Fondecyt 1170313, Conicyt Chile and Conicyt-PCHA/Doc.Nac./2016- 21160684.
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