Open Access

Row stochastic inverse eigenvalue problem

Journal of Inequalities and Applications20112011:24

https://doi.org/10.1186/1029-242X-2011-24

Received: 20 January 2011

Accepted: 21 July 2011

Published: 21 July 2011

Abstract

In this paper, we give sufficient conditions or realizability criteria for the existence of a row stochastic matrix with a given spectrum Λ = {λ1, ..., λ n } = Λ1 Λ m Λm+1, m > 0; where (p k is an integer greater than 1), λk 1= λ k > 0, 1 = λ1ω k > 0, k = 1, ..., m; Λm+1= {λ m +1}, ωm+1λ1 + ..., +λ n λ1, ω k λ k , ω1λ k , k = 2, ..., m + 1. In the case when p1, ..., p m are all equal to 2, Λ becomes a list of 2m + 1 real numbers for any positive integer m, and our result gives sufficient conditions for a list of 2m + 1 real numbers to be realizable by a row stochastic matrix.

AMS classification: 15A18.

Keywords

row stochastic matricesinverse eigenvalue problemrow stochastic inverse eigenvalue problem

1 Introduction and preliminaries

A list of complex numbers Λ is realizable by a matrix A if Λ is the spectrum of A. An n × n nonnegative matrix A = (a mk ) = is a row stochastic matrix if , m = 1, ..., n. The row stochastic inverse eigenvalue problem is the problem of characterizing all possible spectrum of row stochastic matrices. Since row stochastic matrices are important in applications this kind of inverse eigenvalue problems should be interesting. There are also some papers that contribute to the study of the doubly stochastic inverse eigenvalue problem (e.g., [1, 2] and references therein).

In this paper, we give sufficient conditions for some lists of complex numbers, including lists of 2m + 1 real numbers, to be realizable by a row stochastic matrix.

We use to denote the fact that the n × n real square matrix A = (a mk ) satisfies , m = 1, ..., n; use A = diag(A1, ..., A t ) to denote the fact that A is a block diagonal matrix with diagonal blocks A1, ..., A t ; use σ(A) to denote the spectrum of A; use P (n) to denote the permutation matrix of order n the k th row of which is the (k + 1)th row of I n with the first row being (1, 0, ..., 0). Later, we will make use of the fact that the spectrum of ωP(n) is {ω, ωe2πi/n, ωe4πi/n, ..., ωe(n-1)πi/n}.

Since our results are based on the following theorem from [3], we restate it with the proof.

Theorem 1 (Brauer extended) [3] Let A be an n × n arbitrary matrix with eigenvalues λ1, ..., λ n and X = (x1, ..., x t ) be such that rank(X) = t and Ax k = λ k x k , k = 1, ..., t, tn. Let C be a t × n arbitrary matrix. Then the matrix A + XC has eigenvalues μ1, ..., μ t , λt+1, ..., λ n , where μ1, ..., μ t are eigenvalues of the matrix D + CX with D = diag(λ1, ..., λ t ).

Proof Let S = (X, Y ) be a nonsingular matrix with . Then UX = I t , V Y = In-t, VX = UY = 0. Let C = C1, C2), , where both C1, X1 are both t × t and Y1 is t × (n - t). since AX = XD, we have
(1)
Now from ([?]) we have σ(V AY ) = σ(A)\σ(D) and therefore

Lemma 1 If
(2)
(3)
(4)
then the following matrix
(5)

is a row stochastic matrix with eigenvalues λ1, ..., λ t and diagonal entries ω1, ..., ω t .

Proof It is clear that has diagonal entries ω1, ..., ω t and is a nonnegative matrix by (2) and (4). In addition, the eigenpolynomial of B is factorized as

2 Main results

Since the set Λ in Theorem 2 is assumed to be a list of complex numbers, it could be considered as a generalization of Theorem 8 of [3] (Λ in Theorem 8 of [3] is assumed to be a real list). But the representation and the proof of these two theorems almost have no difference.

Theorem 2 [3] Let Λ = {λ1, ..., λ n } be a list of complex numbers. If there exists a partition Λ = Λ1 Λ t , with λ11 = λ1λ21λ31λt 1> 0 and for each Λ k we associate a corresponding list , 0 ≤ ω k ω1 which is realizable by a nonnegative matrix of order p k , k = 1, ..., t, as well as there exists a nonnegative matrix of order t, which has eigenvalues {λ1, λ21, ..., λt 1} and diagonal entries {ω1, ..., ω t }, then Λ is realizable by an n × n nonnegative matrix .

Proof Note the p k -dimensional vector (1, ..., 1)T is an eigenvector of corresponding to the eigenvalue ω k . Let X = (X1, ..., X t ), where X k = (0, ... 0, 1, ..., 1, 0, ..., 0)T is an n-dimensional vector with p k ones from the position p1 + + pk-1+ 1 to p1 + + p k and zeros elsewhere. Let A = diag(A1, ..., A t ), D = diag(ω1, ..., ω t ), then X is of rank t and AX = XD.

Let C = (C1, ..., C t ), where C k is the t × p k matrix whose first column is (c1k, c2k, ..., c tk )T and whose other entries are all zero. Then

where C mk is the p m × p k matrix whose first column is (c mk , c mk , ..., c mk )T and whose other entries are all zero. Now we chose C with c11, ..., c tt = 0 so that the matrix . Then for this choice of C, we conclude that is nonnegative with spectrum Λ by Theorem 1.   

Theorem 3 Let a list of complex numbers Λ = {λ1, ..., λ n } = Λ1 Λ m Λm+1, m > 0; (p k is an integer greater than 1), λk 1= λ k > 0, k = 1, ..., m; Λm+1= {λm+1} be such that 1 = λ1ω k > 0, k = 1, 2, ..., m. Let
(6)
If
(7)
(8)
then Λ is realizable by the following n × n row stochastic matrix
(9)

where M kk = ω k P(p k ), k = 1, ..., m + 1; M kj is the p k × p j matrix whose first column is (ω j - λ j , ..., ω j - λ j )T, kj, j = 2, ..., m + 1 and whose other entries are all zero; Mk 1is the p k × p1 matrix whose first column is (ω1 - λ k , ..., ω1 - λ k )T, k = 2, ..., m + 1 and whose other entries are all zero.

Proof It is clear that is realizable by the nonnegative matrix , k = 1, ..., m + 1. Since , k = 1, ..., m, we have ωm+1= s = λ1 + + λ n = λ1 - ω1 + + λ m - ω m + λm+1by Condition (6) and hence λ1 + + λm+1= ω1 + + ωm+1. Meanwhile if (7) and (8) hold, then all conditions of Lemma 1 are satisfied and hence the row stochastic matrix B defined in (5) with t = m + 1 has eigenvalues {λ1, λ2, ..., λm+1} and diagonal entries {ω1, ..., ωm+1}. Therefore, the list Λ must be realizable by an n × n row stochastic matrix M by Theorem 2. Applying Theorem 2, we compute the solution matrix M and get the result as defined in (9).   

When p k ≤ 2 for all k = 1, ..., m + 1, the set Λ in Theorem 3 becomes a list of real numbers. In this case, applying Theorem 3, we have the following result for real row stochastic inverse eigenvalues problem.

Theorem 4 If a list of real numbers Λ = {λ1, ..., λ2m+1} = Λ1 Λ m {λm+1}; Λ k = {λ k , λ2m+2-k}, k = 1, ..., m satisfies
(10)
(11)
(12)
then Λ is realizable by the following row stochastic matrix
(13)
where

Proof Let μ k = λ k , ω k = -λ2m+2 - k, p k = 2, k = 1, ..., m, ωm+1= s, pm+1= 1, then all the conditions of Theorem 3 are satisfied and Λ is realizable by the row stochastic matrix M defined in (9) by Theorem 3. In the case of Theorem 4, the matrix in (9) becomes the matrix in (13). Therefore, Λ is realizable by the row stochastic matrix M in (13).   

Remark Theorem 10 of [3] gives sufficient conditions only for a list of 5 real numbers to be the spectrum of some 5 × 5 nonnegative matrix ; our Theorem 4 gives sufficient conditions for a list of 2m + 1 real numbers for any integer m > 0 to be the spectrum of some row stochastic matrix. In addition, the conditions of Theorem 4 are more easily handled.

3 Examples

Example 1 Λ = {λ1, ..., λ7} = {1, 0.75, 0.7, 0.1, -0.75, -0.8, -0.8} satisfies Conditions (10), (11) and (12) of Theorem 4 with m = 3, s = 0.2, -λ5 - λ2 = -λ6 - λ3 = 0.05, s - λ4 = 0.1, -λ7 - λ2 = 0.05, -λ7 - λ3 = 0.1, -λ7 - λ4 = 0.7. Therefore Λ is realizable by the following row stochastic matrix:
Example 2 Λ = {λ1, ..., λ6} = {1, 0.2, 0.7e2πi/5, 0.7e4πi/5, 0.7e6πi/5, 0.7e8πi/5} satisfies all the conditions of Theorem 3 with m = 1, p1 = 5, p2 = 1, ω2 = s = 0.5 < 1 = λ1, ω1 = 0.7 > ω2 > 0.2 = λ2, ω2 - λ2 = 0.3, ω1 - λ2 = 0.5. Therefore, Λ is realizable by the following row stochastic matrix:

Declarations

Acknowledgements

We thank the anonymous referee for his/her kind suggestion that leads us to notice the results in [2]. This study was supported by the NSFchina#10871230 and Innovation Group Foundation of Anhui University #KJTD001B.

Authors’ Affiliations

(1)
School of Mathematical Sciences, Anhui University
(2)
School of Sciences, Zhejiang A&F University

References

  1. Yang S, Li X: Inverse eigenvalue problems of 4 × 4 irreducible nonnegative matrices, vol. I. Advances in Matrix Theory and Applications. The proceedings of The Eighth International Conference on Matrix Theory and Applications, World Academic Union 2008.Google Scholar
  2. Kaddoura I, Mourad B: On a conjecture concerning the inverse eigenvalue problem of 4 × 4. Symmetric Doubly Stochastic Matrices. Int Math Forum 2008, 31: 1513–1519.MathSciNetGoogle Scholar
  3. Soto RicardL, Rojo O: Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem. Linear Algebra Appl 2006, 416: 844–856. 10.1016/j.laa.2005.12.026MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Shang-jun and Chang-qing; licensee Springer. 2011

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