Doubly stochastic matrix

We recall that a nonnegative matrix is doubly stochastic if its row sums and column sums are all equal to one. Properties of such matrices were studied in [15]. The following lemma gives some basic properties of such matrices. We omit its easy proofs.

Lemma 3.3.1. Let A be a doubly stochastic matrix. Then

(a) 1 is an eigenvalue of A with corresponding eigenvector [1, ..., 1]^T,

(b) the norm, spectral radius, and numerical radius of A are equal to 1, and (c) A is permutationally similar to a direct sum of irreducible doubly stochastic matrices.

For a matrix A of the form B ⊕ C, we recall the decomposition Γ = Γ1 ∪ Γ2 at the end of Section 2.2, where Γ1 = ∂W (A) \ (∂W (B) ∪ ∂W (C)) and Γ² = ∂W (A) ∩

∂W (B) ∩∂W (C). Based on the properties in the above lemma and Proposition 2.3.3, we are able to determine the value of k(A) for any 3-by-3 doubly stochastic matrix.

Proposition 3.3.2. Let A be a 3-by-3 doubly stochastic matrix. Then k(A) = 3.

Proof. Let A be a 3-by-3 doubly stochastic matrix. Then, by Lemma 3.3.1 (a) and (b), 1 is a reducing eigenvalue of A. This implies that Γ is nonempty. By Proposition

2.3.3, we obtain that k(A) = 3. 

Recall also that the number m of eigenvalues z of A with kzk = r(A) is called the index of imprimitivity of A, and is denoted by m(A). The following result is shown in [15, Corollary 1.5 and Theorem 2.1], which is useful for our later work on a 4-by-4 reducible doubly stochastic matrix.

Proposition 3.3.3. Let A = [aij]³_i,j=1 be a 3-by-3 irreducible doubly stochastic matrix.

(a) If m(A) = 1, then the numerical range W (A) is the convex hull of the point 1 and a compact convex set K contained in the open unit disc D, K is either a (possibly degenerate) elliptic disc with foci (tr A − 1 ± p(tr A − 1)²− 4 det A)/2 ∈ R and minor axis of length √

det A − det Re A, or a (possibly degenerate) elliptic disc with foci (tr A − 1 ±p(tr A − 1)²− 4 det A)/2 ∈ C and minor axis of length (p3|a12− a²¹|²+ (tr A − 1)²− 4 det A)/2, and

(b) if m(A) ≥ 2, then A is normal with the numerical range W (A) the regular 3-polygon with vertices e^2πi/3, 0 ≤ j < 3.

From Proposition 3.3.2, we have proven that for any 3-by-3 doubly stochastic matrix A the value of k(A) is always equal to its size. The following proposition indicates that this still holds for any 4-by-4 reducible doubly stochastic matrix. Note that any reducible doubly stochastic matrix is permutationally similar to a direct sum of irreducible doubly stochastic matrices by Lemma 3.3.1. Applying this result, we have the following proposition.

Proposition 3.3.4. Let A be a 4-by-4 reducible doubly stochastic matrix. Then

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k(A) = 4. Moreover, the following hold:

(a) If A is permutationally similar to a direct sum of two 2-by-2 irreducible dou-bly stochastic matrices H1 and H2, then H1 and H2 are Hermitian, W (A) = [2a − 1, 1], where a = (tr A − p(tr A)²− 4(det A + tr A − 1))/4, with 0 ≤ a < 1, and σ(A) consists of 1, 1, 2a − 1 and 2b − 1, where a is defined above and b = (tr A + p(tr A)² − 4(det A + tr A − 1))/4, with 0 ≤ a ≤ b < 1.

(b) If A is permutationally similar to a direct sum [1] ⊕ B, where B is a 3-by-3 doubly stochastic matrix, then either

(i) B is reducible, W (A) = [tr A − 3, 1], where 2 ≤ tr A ≤ 4, and σ(A) = {1, 1, 1, tr A − 3}, or

(ii) B is irreducible, W (A) = W (B), and σ(A) = {1} ∪ σ(B), both of which were as described in Proposition 3.3.3.

Proof. Since the proof is very similar to Proposition 3.3.3, we omit it. 

The next proposition is concerned with 4-by-4 irreducible doubly stochastic ma-trices. If the index of imprimitivity m(A) equals one, then it shows that A is unitarily similar to a direct sum of [1] and a 3-by-3 matrix B, and the numerical range W (B) is contained in the open unit discD by [15, Theorem 1.2]. Hence we can describe the shape of W (A) in terms of W (B). Note that W (B) has four possible shapes (cf. [9]).

Moreover, if m(A) ≥ 2, then m(A) = 2 or 4 by [13, p. 51].

Proposition 3.3.5. Let A be a 4-by-4 irreducible doubly stochastic matrix.

(a) Assume that m(A) = 1. Then k(A) = 4 if and only if A = [1] ⊕ [λ] ⊕ C, where λ (6= 1) ∈ ∂W (A) ∩ R so that either W (A) is a 4-polygon or W (A) is the convex hull of the (closed) interval [λ, 1] and the elliptic disc W (C).

(b) If m(A) ≥ 2, then k(A) = 4. More precisely, the following hold:

(i) If m(A) = 4, then A is normal with W (A) the regular 4-polygon with vertices e^2πi/4, 0 ≤ j < 4.

(ii) If m(A) = 2, then W (A) is either the closed interval [−1, 1], in which case A is Hermitian with spectrum {±1, ±√

det A}, or the convex hull of the closed interval [−1, 1] and W (B), in which case A is permutationally similar to





0 A1

A2 0



 and W (B) is the elliptic disc with foci ±√

det A and minor axis of length | det A1−det A2|.

Proof. Assume that m(A) = 1. Then A is unitarily similar to the direct sum of [1] and a 3-by-3 matrix B, and the numerical range W (B) is contained in the open unit disc D by [15, Theorem 1.2]. Note that W (A) is symmetric with respect to the x-axis since A is nonnegative. If k(A) = 4 and B is reducible, say, B = C ⊕ [λ], then λ (6= 1) is in ∂W (A) ∩ R and k¹(C) = 2 by Proposition 2.3.8. Hence it follows that W (A) has the asserted shapes. On the other hand, if k(A) = 4 and B is irreducible, then W (B) is the heart-shaped region which is symmetric with respect to the x-axis via A  0. However, this cannot happen by Proposition 2.3.7. Hence we have proven the necessity for k(A) = 4. The converse is trivial.

Assume that m(A) ≥ 2. Then m(A) = 2 or 4 by [13, p. 51]. If m(A) = 4, then case (i) holds trivially. Hence k(A) = 4 by Proposition 2.2.1. On the other hand, if m(A) = 2, then −1 and 1 are eigenvalues of A by the Perron–Frobenius Theorem (cf.

[11, Theorem 15.5.1]). Since −1 and 1 are corners in the boundary of W (A), both are reducing eigenvalues by [3, Theorem 1]. Hence A is unitarily similar to a direct sum of diag (1, −1) and a 2-by-2 matrix B, which shows that W (A) is the convex hull of the closed interval [−1, 1] and W (B). If W (B) is contained in [−1, 1], then it is obvious that B is Hermitian and so is A. In this case, the value k(A) = 4 holds obviously by Proposition 2.2.1. Therefore we may assume that W (B) is not contained in [−1, 1]. This implies that W (B) is an elliptic disc. Furthermore, since W (A) is symmetric with respect to the x-axis, W (B) is also symmetric to the x-axis. Thus W (A) has four line segments, called L1, ..., L4, on its boundary so that L1 is parallel to L2, and L3 is parallel to L4. This shows that k(A) = 4 by [19, Corollary 2.5]. For

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the remaining proof of case (ii), we only need to compute tr A and det A directly.

Hence we complete the proof of case (b). 

Recall that the index of imprimitivity of A is the number of eigenvalues z of A with |z| = r(A). By Lemma 3.3.1, for a doubly stochastic matrix A an eigenvalue with absolute value one is a reducing eigenvalue. This implies that k(A) has the lower bound m(A). Hence we may combine Propositions 3.3.2 and 3.3.4 to give the following result on an n-by-n (n ≥ 3) reducible doubly stochastic matrix.

Theorem 3.3.6. Let A be an n-by-n (n ≥ 3) reducible doubly stochastic matrix.

If n = 3 or 4, then k(A) = n; otherwise, k(A) ≥ max {m(A), 4}.

Our final result is on n-by-n (n ≥ 3) irreducible doubly stochastic matrices.

Theorem 3.3.7. Let A be an n-by-n (n ≥ 3) irreducible doubly stochastic matrix.

(a) If m(A) = 1, then k(A) ≥ 3.

(b) Assume that m(A) ≥ 2.

(i) If n is a prime, then k(A) = n. In this case, A is normal with its numerical range W (A) the regular n-polygon with vertices e^2πi/n, 0 ≤ j < n.

(ii) If n is not a prime, then k(A) ≥ max {m(A), 3}. Moreover, k(A) = m(A) if and only if m(A) ≥ 3, the numerical range W (A) is the m(A)-regular polygon with vertices e^2πi/m(A), 0 ≤ j < m(A), and the dimension of Ha equals 2 for any nonextreme boundary point a of W (A).

Proof. Part (a) is obvious. So we assume that m(A) ≥ 2 in the following. If n is a prime, then A is normal and its numerical range W (A) is the n-polygon with vertices e^2πi/n, 0 ≤ j < n by [15, Corollary 1.5]. Hence k(A) = n by Proposition 2.2.1. This completes the proof of (i). To show (ii), we note that m(A) is exactly the number of reducing eigenvalues of A since A is a doubly stochastic matrix with ω(A) = 1.

This implies that k(A) ≥ m(A). In addition, it is obvious that k(A) ≥ 3 for any

n-by-n (n ≥ 3) doubly stochastic matrix. Hence k(A) ≥ max {m(A), 3}. Assume that is contained in the interior of W (C). This proves the necessity of our assertion. The

convers is obvious. 

We end this section by stating a natural question on k(A) = n for an n-by-n (n ≥ 3) irreducible doubly stochastic matrix A. Is it true that k(A) = n if and only if n ≥ 3, the numerical range W (A) is the n-regular polygon with vertices e^2πi/n, 0 ≤ j < n, and the dimension of H^a equals 2 for any nonextreme boundary point a of W (A)? Obviously, the sufficiency for k(A) = n holds since A must be normal. Nevertheless, the necessity fails. For example, let

A =

Then A is an irreducible doubly stochastic matrix with m(A) = 2 since its spectrum is {1, −1,√

3i/3, −√

3i/3}. In addition, A is clearly not Hermitian. By Proposition 3.3.5 (b) (ii), we have k(A) = 4. However, W (A) is not the regular 4-polygon with vertices e^2πi/4, 0 ≤ j < 4.

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