Linear Algebra and its Applications 439 (2013) 2256–2262
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Comment on “A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices” ✩ Wei-Ru Xu a , Ying-Jie Lei a,∗ , Xian-Ming Gu b , Yong Lu a , Yan-Ru Niu a a
School of Science, North University of China, Taiyuan, 030051, China School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China
b
a r t i c l e
i n f o
Article history: Received 28 November 2012 Accepted 1 June 2013 Available online 18 July 2013 Submitted by S. Kirkland MSC: 15A18 15A51 Keywords: Symmetric positive doubly stochastic matrix Inverse eigenvalue problem Sufficient condition
a b s t r a c t In [S.K. Hwang, S.S. Pyo, The inverse eigenvalue problem for symmetric doubly stochastic matrices, Linear Algebra Appl. 379 (2004) 77–83] it was claimed that: if 1 > λ2 λ3 · · · λn and 1 + n(nλ−2 1) + (n−1λ)(3n−2) + · · · + 2λ·n1 0, then there is a symmetric n positive doubly stochastic matrix A with the eigenvalues 1, λ2 , λ3 , . . . , λn . Afterwards, Fang [M.Z. Fang, A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices, Linear Algebra Appl. 432 (2010) 2925–2927] presented a counterexample to demonstrate that the above proposition was inaccurate. However, the author did not give a solution for a real n-tuple σ = (1, λ2 , λ3 , . . . , λn ) to be the spectrum of a symmetric positive doubly stochastic matrix of order n. In this paper, we give some sufficient conditions to make up for this deficiency. © 2013 Elsevier Inc. All rights reserved.
1. Introduction A real square matrix with nonnegative entries all of whose row sums and column sums are equal to 1 is referred to as doubly stochastic.
✩
*
DOI of original article: http://dx.doi.org/10.1016/j.laa.2009.12.032. Supported by the National Natural Science Foundation of China (Grant No. 11071227). Corresponding author. E-mail addresses:
[email protected] (W.-R. Xu), fl
[email protected] (Y.-J. Lei).
0024-3795/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.06.001
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In particular, a real matrix A = (ai j )n×n is said to be generalized symmetric doubly stochastic if it is symmetric and all its row sums and column sums are a same constant, say α , i.e. n
ai j =
i =1
n
ai j = α ,
i , j = 1, . . . , n .
j =1
The inverse eigenvalue problem for symmetric doubly stochastic matrices is to determine the necessary and sufficient conditions for a real n-tuple σ = (1, λ2 , λ3 , . . . , λn ) to be the spectrum of a n ) the set of all n × n symmetric (symmetric symmetric doubly stochastic matrix. We denote by Ωn (Ω n ) with σ as its spectrum, positive) doubly stochastic matrices. If there exists a matrix A ∈ Ωn ( A ∈ Ω we say that σ is realizable and that A realizes σ . So far, the problem has only been solved for the case n = 3 by Perfect and Mirsky [1]. The case n = 4 for symmetric doubly stochastic matrices of trace zero has also been solved by them in the above reference. But the problem still remains open for the case n 5. The difficulty is that it is very difficult to derive the necessary and sufficient conditions to complete this conundrum, even if many sufficient conditions have appeared. Firstly, the celebrated sufficient condition for the symmetric doubly stochastic inverse eigenvalue problem is the following. Theorem 1. (See [1].) Let n 2 and let σ = (1, λ2 , λ3 , . . . , λn ) be a list of real numbers with 1 λ2 λ3
· · · λn . If 1
λ2 λ3 λn 0 + + ··· + n n(n − 1) (n − 1)(n − 2) 2·1 holds, then the list σ can be realized by a matrix A ∈ Ωn . +
(1)
Martignon [2] and Hwang and Pyo [3] used the same constructive method in Ref. [1] to derive some sufficient conditions for the symmetric doubly stochastic inverse eigenvalue problem respectively. In 2010, Fang [4] presented a list σ = (1, 0, − 23 ) which shows that the following proposition is inaccurate. Proposition 1. (See [3].) Let n 2 and let σ = (1, λ2 , λ3 , . . . , λn ) be a list of real numbers with 1 > λ2 n . λ3 · · · λn . If the inequality (1) holds, then the list σ can be realized by a matrix A ∈ Ω In reference to the above proposition, we give two lists
− 12 , − 23 ).
σ1 = (1, 0, − 25 , − 25 ) and σ2 = (1, 12 ,
They can be realized by the following matrices
⎛
1 1 ⎜9 ⎜ A1 = 20 ⎝ 5 5
9 1 5 5
5 5 1 9
⎞
5 5⎟ ⎟∈Ω 4 9⎠ 1
⎛
and
1 1 ⎜8 ⎜ A2 = 12 ⎝ 2 1
8 1 1 2
2 1 1 8
⎞
1 2⎟ ⎟∈Ω 4 8⎠ 1
respectively. Obviously, neither of the two lists satisfy condition (1) when 1 > λ2 . Thus we propose the following symmetric positive doubly stochastic inverse eigenvalue problem. Problem 1. What are the necessary and sufficient conditions for a real n-tuple to be the spectrum of a symmetric positive doubly stochastic matrix?
σ = (1, λ2 , λ3 , . . . , λn )
Given a list σ = (1, λ2 , λ3 , . . . , λn ) of real numbers with 1 λ2 λ3 · · · λn . From the property of the trace of positive matrices and Perron–Frobenius Theorem, the following conditions
1 + λ2 + λ3 + · · · + λn > 0,
|λi | < 1,
i = 2, 3, . . . , n ,
are necessary for the list σ to be the spectrum of a symmetric positive doubly stochastic matrix. Throughout this paper, I k and J k signify the k × k identity matrix and the k × k matrix all of whose entries are equal to k−1 respectively.
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2. Main results In this section, some sufficient conditions for the symmetric positive doubly stochastic inverse eigenvalue problem will be derived. Firstly, we give the following result. Lemma 1. (See [7].) Let A be an n × n diagonalizable matrix, so that the matrix A can be decomposed into the weighted sum of a series of idempotent matrices A i (i = 1, 2, . . . , n), i.e.
A=
n
ξi A i ,
i =1
n
where A 2i = A i , A i A j = 0 and i =1 A i = I n for any 1 i, j n and i = j. It is a known fact that the spectrum of the matrix A is identical with the sequence {ξ1 , ξ2 , . . . , ξn }.
n ) is diagonalizable, we present the first result as follows. Since each matrix in the set Ωn (Ω Theorem 2. Let n 2 and let
λn > −1. If 1 n
+
λ2 n(n − 1)
+
σ = (1, λ2 , λ3 , . . . , λn ) be a list of real numbers with 1 > λ2 λ3 · · ·
λ3 λn >0 + ··· + (n − 1)(n − 2) 2·1
(2)
n . In particular, if n 3, then the condition (2) is also holds, then the list σ can be realized by a matrix A ∈ Ω necessary. Proof. Firstly, we can construct the following idempotent matrices
A1 = J n , A 2 = I 1 ⊕ J n −1 − J n , A i = I i −1 ⊕ J n − i +1 − I i −2 ⊕ J n − i +2 ,
3 i n.
n It is easy to see that A 2i = A i , A i A j = 0 and i =1 A i = I n for any 1 i, j n and i = j. Then there
n exists a matrix A = (ai j )n×n = A 1 + i =2 λi A i . By Lemma 1, the list σ = (1, λ2 , λ3 , . . . , λn ) is its spec-
trum. Obviously, the matrix A is symmetric and all of its row sums and column sums are equal to 1. Therefore we must prove that the matrix A is positive when the elements of the given list σ satisfy inequality (2). According to [2], we might as well set λ1 = 1 and λn+1 = 0, then all the diagonal and off-diagonal entries of the matrix A are
aii =
i λk − λk+1 k =1
n−k+1
ai j = a ji =
λ 1 − λ2 n
+ λ i +1 , +
i = 1, 2, . . . , n ,
λ2 − λ 3 λ i − λ i +1 + ··· + , n−1 n−i+1
1 i < j n,
respectively. The elements of the given list σ satisfy 1 > λ2 λ3 · · · λn > −1, so a11 a22 · · · an−1,n−1 = ann and ai j = a ji > 0 for all 1 i < j n. From condition (2), we know that
λ2 − λ 3 λn−1 − λn + ··· + + λn n−1 2 λ2 λ3 λn 1 = + > 0. + + ··· + n n(n − 1) (n − 1)(n − 2) 2·1
ann =
1 − λ2 n
+
n . Therefore the matrix A ∈ Ω
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When n 3, the sufficiency of the condition (2) has been proved as above. Next, we will give its necessity. If n = 1 or n = 2, the necessity of the condition (2) obviously holds. The unique realizable matrices are A 1 = (1) and A 2 =
2 1+λ 2 1−λ2 2
1−λ2 2 1+λ2 2
respectively.
If n = 3, we can construct a matrix
⎞
⎛
a b 1−a−b 3 b c 1−b−c ⎠∈Ω A3 = ⎝ 1 − a − b 1 − b − c a + 2b + c − 1
σ = (1, λ2 , λ3 ). Denote by tr( A 3 ) (det( A 3 )) the trace (determinant) of the ma-
with the spectrum trix A 3 , so we have
tr( A 3 ) = 1 + λ2 + λ3 = 2(a + b + c ) − 1 > 0, 2
det( A 3 ) = λ2 λ3 = 3ac − c − 3b + 2b − a. Next, set
1 3
α= +
λ2 6
+
λ3 2
and β =
1 3
+
λ3 6
+
λ2 2
(3) (4)
. Hence, from (3) and (4) we have
2
α + β = (1 + λ2 + λ3 ) > 0, αβ =
1 3
3
(a + 2b + c − 1)(a + c ) + ac > 0.
From the above, it follows that
α , β > 0. Accordingly, the necessity of the condition (2) holds. 2
From this theorem, we can derive the following results. Because the results satisfy Theorem 2, we omit the proofs. Corollary 1. Given a real n-tuple σ = (1, λ2 , λ3 , . . . , λn ) with 1 > λ2 λ3 · · · λn 0, then there exists n to realize it. a matrix A ∈ Ω Corollary 2. Given a real n-tuple min2i n λi >
1−max2i n λi − n
σ = (1, λ2 , λ3 , . . . , λn ) with λi ∈ (− n−1 1 , 1) for any 2 i n. If
n to realize it. , then there exists a matrix A ∈ Ω
The two lists σ1 and σ2 in Section 1 don’t satisfy condition (2), but they can be realized. Next, we present the constructive method. Lemma 2. Let A ( B ) be an m × m generalized symmetric doubly stochastic matrix all of whose row sums and column sums are equal to s (1 − s) and let
σ ( A ) = (s, ξ2 , ξ3 , . . . , ξm ), σ ( B ) = (1 − s, η2 , η3 , . . . , ηm ) be the spectra of the matrices A and B respectively. If A B = B A, then the spectrum of the matrix
D=
A B
B A
(5)
is σ ( D ) = (1, 2s − 1, ξ2 ± η2 , ξ3 ± η3 , . . . , ξm ± ηm ). Proof. Because the matrices A and B are generalized symmetric doubly stochastic and A B = B A, there exists an m × m orthogonal matrix U such that the two matrices are simultaneously diagonalizable (see Ref. [5]). Without loss of generality, we take the first column of the orthogonal matrix U to be ( √1 , √1 , . . . , √1 )T . Therefore, we have m
m
m
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U T AU = Λ A = diag{s, ξ2 , ξ3 , . . . , ξm }, U T BU = Λ B = diag{1 − s, η2 , η3 , . . . , ηm }. From the properties of the block matrices in the matrix D, we know that the matrix D is also a generalized symmetric doubly stochastic matrix all of whose row sums and column sums are equal to 1. Therefore, we can construct a 2m × 2m orthogonal matrix
1 R=√ 2
U U
U −U
such that the matrix D is diagonalizable, i.e.
T
R DR =
1
2
=
UT
UT
UT
−U T
A B
B A
U U
U −U
=
U T ( A + B )U
0
0
U T ( A − B )U
Λ A + ΛB
0
0
Λ A − ΛB
Then the spectrum of the matrix D is
.
σ ( D ) = (1, 2s − 1, ξ2 ± η2 , ξ3 ± η3 , . . . , ξm ± ηm ). 2
Furthermore, we can derive the following corollary. Corollary 3. If the list (1, 2s − 1, ξ2 ± η2 , ξ3 ± η3 , . . . , ξm ± ηm ) as above can be realized by a matrix A ∈ Ω2m t and s = 2− for any 0 t 2, then the list (1 + t , 1 − t , ξ2 ± η2 , ξ3 ± η3 , . . . , ξm ± ηm ) can be realized by a 2 nonnegative matrix. In particular, if 0 < t 2, then the list (1 + t , 1 − t , ξ2 ± η2 , ξ3 ± η3 , . . . , ξm ± ηm ) can be also realized by a positive matrix. Proof. Let e be the 2m-dimensional vector of all ones and let e1 be the first column of the 2m × 2m identity matrix. If 0 t 2, then we can construct a 2m × 2m nonnegative matrix A + teeT1 to realize the list (1 + t , 1 − t , ξ2 ± η2 , ξ3 ± η3 , . . . , ξm ± ηm ). If 0 < t 2, then we can construct a 2m × 2m positive matrix A + euT to realize the list (1 + t ,
2m 1 − t , ξ2 ± η2 , ξ3 ± η3 , . . . , ξm ± ηm ), where u = (t 1 , t 2 , . . . , t 2m )T with t = i =1 t i for any t i > 0. 2 An immediate consequence of Lemma 2 is the following. Theorem 3. Let n = 2m (m 2) for m ∈ Z and let 1 > λ2 λ3 · · · λn > −1. If
1 + λm+1 m
+
σ = (1, λ2 , λ3 , . . . , λn ) be a list of real numbers with
λ3 + λm+3 λm + λ2m λ2 + λm+2 + + ··· + >0 m(m − 1) (m − 1)(m − 2) 2·1
(6)
and
1 − λ2 − λm+1 + λm+2 > 0,
λi − λi +1 − λm+i + λm+i +1 0, i = 2, . . . , m − 1, n . hold, then the list σ can be realized by a matrix A ∈ Ω
(7)
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Proof. Firstly, select an m × m Soules matrix
⎛
√1
1 m(m−1) √ 1 m(m−1) √ 1 m(m−1)
√
m √1 m √1 m
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ . U = (u i j )m×m = ⎜ .. ⎜ ⎜ 1 ⎜√ ⎜ m ⎜ 1 ⎜√ ⎝ m √1
m
√ √ √
1
1 m(m−1) √ 1 m(m−1) 1 √ − mm(− m −1 )
√1 4×3
√1 3×2
···
√1 4×3
√1 3×2
···
√1 4×3
− √32×2
0
.. .
..
.
.. .
.. .
.. .
1
···
0
0
0
···
0
0
0
···
0
0
0
1
(m−1)(m−2) 1
(m−1)(m−2)
.. . √
···
(m−1)(m−2)
√
(m−1)(m−2) − √(m−m1−)(2m−2)
0
in [6], where u i j = 0 for i + j > m + 2, u i j = − √ i −1
i (i −1)
1 √ (m− j +2)(m− j +1)
√1 2×1
−√ 1
2×1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
for i + j = m + 2 and u i1 = √1 , u i j = m
( j = 1) for i + j < m + 2. Because of n = 2m (m 2) for m ∈ Z, then from Lemma 2 we can construct an n × n U U orthogonal matrix R = √1 U −U to derive a matrix C = R T Λ R of the form (5), where Λ = 2 diag{1, λ2 , λ3 , . . . , λn } with 1 > λ2 λ3 · · · λn > −1. Next, decompose the diagonal matrix Λ into the direct sum of the two diagonal matrices Λ1 and Λ2 , where Λ1 = diag{1, λ2 , . . . , λm } and Λ2 = diag{λm+1 , λm+2 , . . . , λ2m }. Then we can obtain the following formula: T
C = (c i j )n×n = R Λ R =
=
1
2
1
2
UT
UT
UT
−U T
U T (Λ1 + Λ2 )U U T (Λ1 − Λ2 )U U T (Λ1 − Λ2 )U U T (Λ1 + Λ2 )U
Λ1
=
1 2
Λ2
C1 C2
U U
C2 C1
U −U
.
From Lemma 2, we know that the matrix C is a generalized symmetric doubly stochastic matrix all of whose row sums and column sums are equal to 1. Now, we prove that it is a positive matrix under the circumstance of conditions (6) and (7). Equivalently, it only needs us to analyze the positivity of the matrices C 1 and C 2 . From the matrix C 1 we know that: 1 (1) Because all the off-diagonal entries of the matrix C 1 involve m (1 − λ2 + λm+1 − λm+2 ) and all the other terms involve sums of nonnegative terms, they are all positive. (2) The diagonal entries of the matrix C 1 satisfy cmm cm−1,m−1 · · · c 33 c 22 = c 11 and c 11 = 1+λm+1 m
+
λ2 +λm+2 m(m−1)
are positive.
+
λ3 +λm+3 (m−1)(m−2)
+ ··· +
λm +λ2m . Then from condition (6), all the diagonal entries 2·1
Hence, the matrix C 1 is positive. In addition, from the matrix C 2 we know that: 1 (1) Because all the diagonal entries of the matrix C 2 involve m (1 − λm+1 ) and all the other terms involve sums of nonnegative terms, all the diagonal entries are positive. 1 (2) c in = cni = m (1 − λ2 − λm+1 + λm+2 ) for any i = 1, 2, . . . , m − 1 and all the other off-diagonal
entries of the matrix C 2 not only involve
1 (1 − λ2 m
− λm+1 + λm+2 ), but also involve some of the
terms m−1i +1 (λi − λi +1 − λm+i + λm+i +1 ) for all i = 2, 3, . . . , m − 1. Then from condition (7), they are positive. Hence, the matrix C 2 is also positive.
n ) and the list Consequently, the matrix C is positive (i.e. C ∈ Ω proof is completed. 2
σ can be realized by it. So the
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In conclusion, for Problem 1, while there are still many sufficient conditions for us to explore, we have improved the consequence of the paper [4]. Acknowledgements The authors are grateful to the referees for their valuable comments and suggestions which helped to improve the presentation of this paper. References [1] H. Perfect, L. Mirsky, Spectral properties of doubly-stochastic matrices, Monatsh. Math. 69 (1965) 35–37. [2] L.F. Martignon, Doubly stochastic matrices with prescribed positive spectrum, Linear Algebra Appl. 61 (1984) 11–13. [3] S.K. Hwang, S.S. Pyo, The inverse eigenvalue problem for symmetric doubly stochastic matrices, Linear Algebra Appl. 379 (2004) 77–83. [4] M.Z. Fang, A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices, Linear Algebra Appl. 432 (2010) 2925–2927. [5] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, 1985. [6] G.W. Soules, Constructing symmetric nonnegative matrices, Linear Multilinear Algebra 13 (1983) 241–251. [7] T.-Z. Huang, S.-M. Zhong, Z.-L. Li, Matrix Theory, Higher Education Press, Beijing, 2003 (in Chinese).
