Row stochastic inverse eigenvalue problem

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 ∪ Λ_m+1, m > 0; where (p _k is an integer greater than 1), λ_{k 1}= λ _k > 0, 1 = λ₁ ≥ ω _k > 0, k = 1, ..., m; Λ_m+1= {λ _m +1}, ω_m+1≡ λ₁ + ..., +λ _n ≤ λ₁, ω _k ≥ λ _k , ω₁ ≥ λ _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.

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, ωe^2πi/n, ωe^4πi/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, t ≤ n. Let C be a t × n arbitrary matrix. Then the matrix A + XC has eigenvalues μ₁, ..., μ _t , λ_t+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 = I_n-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 = λ₁ ≥ λ₂₁ ≥ λ₃₁ ≥ ⋯ ≥ λ_{t 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, ..., λ_{t 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 + ⋯ + p_k-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 (c_1k, c_2k, ..., 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 ∪ Λ_m+1, m > 0; (p _k is an integer greater than 1), λ_{k 1}= λ _k > 0, k = 1, ..., m; Λ_m+1= {λ_m+1} be such that 1 = λ₁ ≥ ω _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, k ≠ j, j = 2, ..., m + 1 and whose other entries are all zero; M_{k 1}is 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 ω_m+1= s = λ₁ + ⋯ + λ _n = λ₁ - ω₁ + ⋯ + λ _m - ω _m + λ_m+1by Condition (6) and hence λ₁ + ⋯ + λ_m+1= ω₁ + ⋯ + ω_m+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, ..., λ_m+1} and diagonal entries {ω₁, ..., ω_m+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, ..., λ_2m+1} = Λ₁∪ ⋯ ∪Λ _m ∪{λ_m+1}; Λ _k = {λ _k , λ_2m+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 = -λ_{2m+2 - k}, p _k = 2, k = 1, ..., m, ω_m+1= s, p_m+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, 0.7e^2πi/5, 0.7e^4πi/5, 0.7e^6πi/5, 0.7e^8πi/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: