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Frobenius Normal Forms of Doubly Stochastic Matrices Received August 15, 2019; Accepted November 22, 2019

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Spec. Matrices 2019; 7:213–217 Note Open Access Special Issue Dedicated to Charles R. Johnson Pietro Paparella* Frobenius normal forms of doubly stochastic matrices https://doi.org/10.1515/spma-2019-0015 Received August 15, 2019; accepted November 22, 2019 Abstract: An elementary proof of a fundamental result on doubly stochastic matrices in Frobenius normal form is given. This result is used to establish several well-known results concerning permutations, including a theorem due to Runi. Keywords: Frobenius normal form, doubly stochastic matrix, permutation, permutation matrix MSC: 15A21; 05A05 1 Introduction In 1965, Perfect and Mirsky [5, Lemma 3] stated, without proof, that a doubly stochastic matrix is permuta- tionally similar to a direct sum of irreducible, doubly stochastic matrices. Furthermore, they state that the result “is almost certainly well-known” and omit a proof because it “follows very easily from the denitions” [5, p. 38]. This result is fundamental in the Perfect–Mirsky conjecture, but, to the best of our knowledge, it ap- pears sparingly in the literature: Liu and Lai [4, Theorem 2.7.4] prove the weaker result that a doubly stochastic matrix is either irreducible or permutationally similar to a direct sum of two doubly stochastic matrices; and Hartel and Spellman [2, Lemma 1(b)] give a proof via strong induction. In this work, an elementary proof of this fundamental result is provided that relies on weak induction and the Frobenius normal form of a matrix. To demonstrate its import and utility, we apply it to permutation matrices and and characterize the Frobenius normal forms of a permutation matrix. This, in turn, is used to derive two well-known results concerning permutations, including the disjoint cyclic form and the result due to Runi that the order of a permutation in disjoint cyclic form is the least common multiple of the lengths of its disjoint cycles. 2 Background n For F = C or F = R, we let Mn(F) denote the set of n-by-n matrices with entries over F, and F denote the collection of all column vectors of length n over F. We let In denote the n-by-n identity matrix and e denotes the all-ones vector (the size of which is determined by the context in which it appears). Finally, for n 2 N, we let hni := f1, ... , ng. A directed graph (or simply digraph) Γ = (V, E) consists of a nite, nonempty set V of vertices, together with a set of arcs E ⊆ V × V. A digraph Γ is strongly connected if, for any two vertices u and v of V, there is a (directed) walk in Γ from u to v Every vertex of V is considered strongly connected to itself so that strong *Corresponding Author: Pietro Paparella: University of Washington Bothell, E-mail: [email protected] Open Access. © 2020 Pietro Paparella, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. 214 Ë Pietro Paparella connectivity denes an equivalence relation, and hence a partition, of the vertices into strongly connected components. Notice that Γ is strongly connected if and only if Γ possesses one strongly connected component. If A 2 Mn(C), then the digraph of A, denoted by Γ(A), has vertices V = hni and arcs E = f(i, j) 2 V × V | aij ≠ 0g. For n ≥ 2, a matrix A 2 Mn(C), is reducible if there is a permutation matrix P such that " # > A A P AP = 11 12 , 0 A22 in which A11 and A22 are nonempty square matrices and 0 is a zero block. If A is not reducible, then A is called irreducible. It is well-known that A is irreducible if and only if Γ (A) is strongly connected (see, e.g., Brualdi and Ryser [1, Theorem 3.2.1]). If A 2 Mn(C), then there is a permutation matrix P such that 2 3 A11 ··· A1k > 6 7 P AP = 6 .. 7 , 4 . 5 Akk > in which the matrices A11, ... , Akk are irreducible [1, Theorem 3.2.4]. The matrix P AP is called a Frobe- nius normal form (FNF) of A and is not unique. The irreducible matrices A11, ... , Akk, called the irreducible components of A, are unique up to permutation similarity. 3 Main Result > Recall that a nonnegative matrix A 2 Mn(R) is called doubly stochastic if Ae = e = A e. With this denition, we are now ready to present the main result. Theorem 3.1. Let A 2 Mn(R) and suppose that 2 3 A11 ··· A1k > 6 7 P AP = 6 .. 7 4 . 5 Akk is a Frobenius normal form of A. If A is doubly stochastic, then Aij = 0, i < j, i.e., 2 3 k A11 > M 6 7 P AP = Aii = 6 .. 7 4 . 5 i=1 Akk and the irreducible components A11 2 Mn1 (R), ... , Akk 2 Mnk (R) are doubly stochastic. Proof. Proceed by induction on k, the number of irreducible blocks in any FNF of A. When k = 1, then A is irreducible and the result is clear. For the induction-step, assume that k ≥ 2 and that the result holds for any doubly stochastic matrix having k − 1 irreducible blocks in any FNF. > > > Since Pe = P e = e, it follows that P AP is doubly stochastic. Consequently, A11e = e and A11e ≤ e (inequality here is considered entrywise). We claim that A11e = e. Otherwise, > > > > n1 = e e = e A11 e = e (A11e) < e e = n1, Spectrally Perron Polynomials Ë 215 a contradiction. Thus, A11 is doubly stochastic and 2 3 A11 6 A A 7 > 6 22 ··· 2k7 P AP = 6 7 . 6 .. 7 4 . 5 Akk The submatrix 2 3 A22 ··· A2k 6 7 6 .. 7 4 . 5 Akk > of P AP is doubly stochastic and has k − 1 irreducible blocks; as such, the induction-hypothesis applies and the result is established. n > n Remark 3.2. For α > 0 let CSα := fA 2 Mn(R) | A ≥ 0, Ae = αe = A eg. Notice that A 2 CSα if and only if n n ri = cj = α, 8i, j 2 hni. Furthermore, A 2 CSα if and only if B := A/α 2 CS1, therefore Theorem 3.1 applies to n matrices in CSα, α > 0. 4 Permutations and Permutation Matrices. Denote by Sn the symmetric group of hni. For σ 2 Sn, the permutation matrix corresponding to σ, denoted by Pσ, is the the n-by-n matrix such that (i, j)-entry is δi,σ−1(i), where δij denotes the Kronecker delta. When the context is clear, Pσ is abbreviated to P. As is well-known, Pσ Pγ = Pσγ and so the set of all n-by-n permutation matrices, denoted by Pn, forms a group under matrix multiplication and the map Φ : Sn −! Pn, dened by Φ(σ) = Pσ is an isomorphism. Observation 4.1. If P 2 Mn(R) is a permutation matrix, then P is irreducible if and only if Γ (P) is an n-cycle. Proof. Let v 2 hni. Since Γ (P) is strongly connected, Γ (P) has a k-cycle of distinct vertices v := v1 −! ··· −! vk −! v1 with 1 ≤ k ≤ n. We claim that k = n; otherwise, if k < n, then Γ (P) clearly contains more than one connected component, which is a contradiction. Thus, k = n, i.e., Γ (P) is an n-cycle. Conversely, the matrix P is clearly irreducible when Γ (P) is an n-cycle. " # 0 1 Observation 4.2. For a positive integer n ≥ 2, the matrix Cn := is called a basic circulant. No- In−1 0 tice that Cn is the permutation matrix corresponding to the cyclic permutation 1 ··· n . Thus, if P is an > irreducible permutation matrix, then there is a permutation matrix Q such that Q PQ = Cn. Corollary 4.3 (FNF of a permuation matrix). If P is a permutation matrix, then there is a permutation matrix Q Q>PQ Lk C k n such that = i=1 ni , 1 ≤ ≤ . Lemma 4.4. If α = α1 ··· αk 2 Sn and β = β1 ··· β` 2 Sn are disjoint, cyclic permutations (i.e., −1 −1 αi ≠ βj) and γ 2 Sn, then γαγ and γβγ are disjoint, cyclic permutations. −1 Proof. If yi := γ(αi), 1 ≤ i ≤ k, then γ(α(γ (yi))) = γ(α(αi)) = γ(αi+1) = yi+1, where, for convenience, −1 −1 −1 αk := α and yk := y . If y 2 hninfy , ... , ykg and x := γ (y), then (γαγ )(y) = γ(α(γ (y))) = γ(α(x)) = +1 1 +1 1 1 −1 −1 γ(x) = y. Thus, γαγ = y1 ··· yk . A similar argument demonstrates that γβγ = z1 ··· z` with zi := γ(βi), 1 ≤ i ≤ `. 216 Ë Pietro Paparella For contradiction, if γαγ−1 and γβγ−1 are not disjoint, then there are positive integers i and j such that γ(αi) = yi = zj = γ(βj), i.e., αi = βj, a contradiction. Theorem 4.5 (disjoint cyclic form). If σ 2 Sn, then there are disjoint, cyclic permutations σ1, ... , σk 2 Sn, k n σ Qk σ 1 ≤ ≤ , such that = i=1 i. Proof. If P is the permutation matrix corresponding to σ, then, following Corollary 4.3, there is a permutation matrix Q corresponding to γ 2 Sn such that k ! k M > M > P = Q Cni Q = QCni Q , 1 ≤ k ≤ n. i=1 i=1 > If σˆi is the cyclic permutation corresponding to Cni and σi is the permutation corresponding to QCni Q , i.e., σ γσ γ−1 σ ... σ σ Qk σ i = ˆi , then 1, , k are pairwise disjoint (Lemma 4.4) and = i=1 i, as desired.
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