Classification of k-primitive sets of matrices

Classification of k-primitive sets of matrices V. Yu. Protasov

∗

†

Abstract We develop a new approach for characterizing k-primitive matrix families. Such families generalize the notion of a primitive matrix. They have been intensively studied in the recent literature due to applications to Markov chains, linear dynamical systems, and graph theory. We prove, under some mild assumptions, that a set of k nonnegative matrices is either k-primitive or there exists a nontrivial partition of the set of basis vectors, on which these matrices act as commuting permutations. This gives a convenient classification of k-primitive families and a polynomial-time algorithm to recognize them. This also extends some results of Perron-Frobenius theory to nonnegative matrix families. Keywords: nonnegative matrix, primitivity, Hurwitz products, partition, permutation, polynomial algorithm AMS 2010 subject classification: 15B48, 05A05, 20B20

I. Introduction The notion of primitivity of a nonnegative matrix plays a crucial role in the PerronFrobenius theory. Special properties of primitive matrices, such as convergence of powers, the simplicity of the largest eigenvalue, projective contraction on the positive orthant, etc., have been put to good use in applications. There are several ways to extend this concept to families of nonnegative matrices, possibly, not commuting (see [4, 15, 16, 19] and references therein). In this paper we deal with k-primitive families introduced by Fornasini and Valcher in [9] and studied in [2, 10, 11, 14, 17, 21]. We present a new method to characterize and classify such families and show that the k-primitivity can be determined in polynomial time. Let us have a finite family of nonnegative d × d-matrices A = {A1 , . . . , Ak }. For a given ∑ k-tuple α = (α1 , . . . , αk ) of nonnegative integers, ki=1 αi = m ≥ 1, we denote by Aα the ∗

The research is supported by RFBR grants nos. 11-01-00329 and 13-01-00642, by the program “Leading Scientific Schools” (grant no. NSh-6003.2012.1), and by the Dynasty foundation † Dept. of Mechanics and Mathematics, Moscow State University, Vorobyovy Gory, 119992, Moscow, e-mail: [email protected]

1

sum of all products of m matrices from A, in which every product contains exactly αi factors equal to Ai . The number m will be referred to as the length of α and denoted by |α|. For example, if A = {A1 , A2 , A3 }, then A (1,3,0) = A1 A32 + A2 A1 A22 + A22 A1 A2 + A32 A1 . Such sums are called in the literature Hurwitz products (see, for instance, [7]). They naturally appear as coefficients of homogeneous matrix polynomials: ∑ ( )m tt A1 + · · · + tk Ak = tα Aα , |α|=m

where t α = t1α1 · · · t1αk . Definition 1 [9]. A family A of k nonnegative matrices is called k-primitive, if there exists a k-tuple α such that the matrix A α is entrywise strictly positive. This notion has an obvious combinatorial interpretation. Consider a directed multigraph with d vertices, its arcs are colored with k colors. The k-primitivity means that there exists a k-tuple α = (α1 , . . . , αk ) such that every two vertices are connected by a path of length |α| that for each i = 1, . . . , k contains precisely αi arcs of the ith color. The concept of k-primitivity is applied in the study of inhomogeneous Markov chains, multivariate generalizations of Markov chains (see [8] and references therein), linear switching systems, in particular, 2D systems [9, 11, 17], graphs and multigraphs [3, 2, 13, 14], etc. In case of one matrix (k = 1) this notion reduces to usual primitivity. A nonnegative matrix A is called primitive if AN > 0 for some N . According to the classical result of Wielandt, the minimal power N = N (A) with this property never exceeds (d − 1)2 + 1, where d is the matrix dimension. The Perron-Frobenius theory suggests several methods to verify the primitivity of one matrix (see, for instance [12, chapter 8]). However, for a set of matrices A1 , . . . , Ak the situation is more difficult: instead of working with powers of one matrix, we have to deal with nonhomogeneous matrix products. The main result in this direction obtained in the literature provides an algebraic criterion of k-primitivity in terms of cycles of the corresponding multigraphs [9, 17, 21]. To every family A of k nonnegative d × d-matrices we associate a directed multicolored graph (multigraph) G. It has d vertices numbered from 1 to d, each arc is colored in one of k colors. There is an arc of color s from a vertex i to a vertex j if and only if (As )ji > 0. For a given nonnegative vector x ∈ Rd , we call the support of x the set of indices of its strictly positive entries. The zero vector has empty support. A family A is called reducible if all its matrices share a common invariant coordinate subspace. This means that there is a proper nonempty subset Λ ⊂ {1, . . . , d} such that for each As ∈ A and for every i ∈ Λ the support of the ith column of As is contained in Λ. A family is irreducible if and only if its multigraph G is strongly connected, i.e., for every pair of vertices a, b there is a path leading from a to b. Reducibility is equivalent to the existence of a suitable permutation of the basis of Rd , after which all matrices from A have a block upper-triangular form. Hence, every product of those matrices has the same form. Therefore, a reducible family is never k-primitive. The problem becomes to characterize k-primitivity of irreducible families. The algebraic criterion for that was presented in [9] (for k = 2) and then in [21] and in [17] (for general k). To 2

formulate it we need to introduce some more notation. By |K| we denote the cardinality of a finite set K. For every finite path γ on the multigraph G, we write (γ) for the k-tuple (γ1 , . . . , γk ), where γi is the total number of arcs of ith color∑ in this path. We call (γ) the color vector of the path γ. The length of γ is equal to |(γ)| = ki=1 γi . An empty path (from a vertex to itself) has length zero, its color vector is a zero vector. A closed path with no repeated vertices except for the first and last is called cycle. An empty path is a cycle. Let c1 , . . . , cr be all cycles of G. Consider the set M

=

r {∑

ni · (ci ) ni ∈ Z ,

} i = 1, . . . , r

.

(1)

i=1

This is an integer sublattice of Zk generated by color vectors of all cycles. Since each closed path is a sum of several cycles, the lattice M contains color vectors of all closed paths of G. Theorem A [9, 21, 17]. An irreducible family of nonzero matrices is k-primitive if and only if M = Zk . Thus, A is k-primitive if k-tuples of cycles of the multigraph G generate the entire Zk . This is equivalent to say that ind(M ) = 1, where the index of lattice M is the g.c.d. of determinants of all possible k × k-matrices composed by the generators (ci ) , i = 1, . . . , r. Hence, Theorem A generalizes a well-known fact of Perron-Frobenius theory, also called Romanovsky’s theorem (see [20]): a nonnegative irreducible matrix is primitive if and only if the g.c.d. of lengths of all cycles of the corresponding digraph is 1. In practice, the use of the criterion of Theorem A to check the k-primitivity may be difficult, especially for high values of d. To ensure the k-primitivity one needs to show that the lattice generated in Zk by all cycles of the digraph G coincides with Zk . It is a priory not clear how to choose cycles for that; on the other hand, the exhaustion of all cycles may be too expensive, because their total number may be exponential in d. A question arises: can k-primitivity be solved by an efficient polynomial algorithm ? The results from [17] lead to rather pessimistic conclusions: it was shown that there are k-primitive families A, for which the minimal length of k-tuples α such that A α > 0 grows as d k+1 , i.e., exponentially in k. Whence, the naive algorithm of checking Hurwitz products of some given length (like in the case k = 1) is too expensive. In this paper we suggest a new approach to classify k-primitive families (Theorem 1). This leads, under some mild assumptions, to a polynomial-time algorithm for determining k-primitivity. The complexity of the algorithm is O(kd 3 + k 2 d 2 ) arithmetic operations. We briefly describe and discuss it here, and give precise formulations in the next section. Throughout the paper we denote by Ω the set {1, . . . , d}, where d is the matrix size. Consider an arbitrary partition of the set Ω into n nonempty pairwise disjoint sets Ω1 , . . . , Ωn . The partition is nontrivial if n ≥ 2. The set of elements of the partition will be denoted by Q. Thus, Q = {Ω1 , . . . , Ωn }. We say that a nonnegative matrix A acts as a permutation on this partition, if there is a permutation f of the set Q such that for every i ∈ Ωj the support of the ith column of A is contained in f (Ωj ). Thus, the matrix A sends all basis vectors with indices from Ωj to a subspace spanned by basis vectors with indices from f (Ωj ). Permutations f1 and f2 commute if f1 f2 (Ωj ) = f2 f1 (Ωj ) for all Ωj ∈ Q. 3

Thus, a family A = {A1 , . . . , Ak } of nonnegative d × d-matrices act as commuting permutations on a given partition Ω = ∪ni=1 Ωi , if there are permutations f1 , . . . , fk of the set Q = {Ω1 , . . . , Ωm } that commute with each other (fi fj = fj fi for every i, j ∈ {1, . . . , k}) and such that every matrix Am ∈ A acts as the permutation fm . Theorem 1 formulated in the next section gives the following criterion of k-primitivity. Suppose that either no matrix of A has a row of zeros or no matrix of A has a column of zeros; then A is not k-primitive if and only if there is a nontrivial partition Ω = ∪ni=1 Ωi , on which all matrices of A act as commuting permutations. This fact has the following interpretation. A family A is not k-primitive precisely when there exists a permutation of the basis of Rd , after which all matrices A ∈ A have a block form of the type:   0 0 B1 0 ... 0  0 0 0 0 . . . B2     .. .. .. .. ..   . . . .  (2) A =  .   Bn−2 0 0 0 ... 0     0 0 0 Bn−1 . . . 0  0 Bn 0 0 ... 0 with n2 rectangular blocks corresponding to the sets Ωi . All but n of these blocks are zero. The matrices Bm and their positions depend on A. The sizes of blocks are |Ωi | × |Ωj | , i, j = 1, . . . , n. Each of the n columns consisting of n blocks has only one nonzero block. The same holds for rows of block: each row has only one nonzero block. There are no special assumptions on the matrices B1 , . . . , Bn , they are arbitrary nonnegative matrices (of the corresponding sizes). However, if the matrix A has no row of zeros (or no columns of zeros), then so does each matrix Bm . The partition to blocks is the same for all matrices from A, but different matrices may have different positions of nonzero blocks. Moreover, the block structures of every two matrices As , At ∈ A commute, i.e., all nonzero blocks of the matrix As At are located on the same positions as nonzero blocks of the matrix At As . Example 3 in the next section will show that the the nonzero rows (or nonzero columns) assumption in Theorem 1 cannot be omitted. On the other hand, this assumption is not very restrictive in applications. For instance, in the study of 2D Markov chains we deal with pairs of matrices A = {A1 , A2 } of the form A1 = aP1 , A2 = (1 − a)P2 , where P1 , P2 are stochastic matrices and 0 ≤ a ≤ 1 [8, theorem 2.1]. Therefore, apart from the trivial cases a = 0, a = 1, the pair A satisfies this assumption. Theorem 1 can be efficiently applied to characterize k-primitive families. We prove that the partition of the set Ω, on which all matrices from A act as commuting permutations, can be found explicitly by a polynomial-time procedure (Theorem 2). This is the base of the algorithm for deciding k-primitivity. We present it in Section V. Let us remark that Theorem 1 generalizes another fact of the Perron-Frobenius theory: if an irreducible matrix A is not primitive, then there is a partition of the set Ω, on which A acts as a cyclic permutation [12, chapter 8]. Theorem 1 extends this to families of matrices, but the permutations are already not necessarily cyclic. Moreover, among all partitions Ω = ∪ni=1 Ωi on which A acts as commuting permutations there is a unique partition of maximal 4

cardinality n (Proposition 1), and this number n plays a role of the imprimitivity index (see Remark 2 for details). It is also interesting to observe relations between different concepts of primitivity of matrix families studied in [4, 18, 19]. We discuss this issue in Remark 4, Section II, after formulating the main results. In Sections III we analyze several examples and then, in Sections IV and V, give proofs. II. The main results Let A = {A1 , . . . , Ak } be an arbitrary family of nonnegative d × d-matrices. We consider two conditions on this set: (a) The family A is irreducible. (b) Either no matrix of A has a row of zeros or no matrix of A has a column of zeros. The following theorem gives a criterion of k-primitivity. Theorem 1 A family A satisfying (a) and (b) is not k-primitive if and only if there exists a nontrivial partition Ω = ∪ni=1 Ωi of the set Ω = {1, . . . , d}, on which all matrices from A act as commuting permutations. Actually, the “if” part of the criterion is obvious. Moreover, it holds for every matrix family, possibly not satisfying conditions (a) and (b). Indeed, if all matrices from A act as permutations of some partition, then they have block form (2). Since these permutations commute, it follows that all matrix products in the sum A α have the same location of nonzero blocks. Thus, for every k-tuple α, the matrix A α has the form (2), and hence it cannot be totally positive. The proof of the “only if” part is more complicated, we give it in Section IV. Remark 1 As we mentioned above, the irreducibility assumption (a) is significant in Theorem 1, because a reducible family is never k-primitive. Assumption (b) cannot be omitted either. Example 3 in the next section shows the situation, when a pair of 6 × 6-matrices is not k-primitive, although there is no partition, on which both matrices act as commuting permutations. The reason is that those matrices possess zero rows and columns. We are not aware of any simpler example in a lower dimension. Note that the partition from Theorem 1 may not be unique. For example, if d = 4, A is the matrix of cyclic permutation of the set Ω = {1, 2, 3, 4}, then, for the family A = {A}, there are two partitions: Ω = ∪4i=1 {i} and Ω = {1, 3} ∪ {2, 4}. For a given family A, consider the partition Ω = ∪ni=1 Ωi from Theorem 1 (if there are several ones, take one of them). Each matrix As ∈ A defines a permutation of the set Q = {Ωi }ni=1 . We denote this permutation by the same symbol As . The group F generated by those k permutations is a finite abelian group that acts on the set Q. By the irreducibility assumption, this action is transitive. Take now the lattice M ⊂ Zk defined in (1) and consider the corresponding additive quotient group Zk |M . Let us recall that ind (M ) denotes the index of the lattice. 5

Proposition 1 For a given family A, among all partitions Ω = ∪ni=1 Ωi from Theorem 1 there exists a unique one with the maximal number of elements n. For this partition, we have F ∼ = Zk |M and n = |F | = ind (M ). For every other partition, the group of permutations is a proper subgroup of F , and the number of elements of the partition is a divisor of n. The proof is given in Section IV. This unique partition will be referred to as maximal, and the number n of its elements is the k-imprimitivity index of the family A. By Proposition 1, the k-imprimitivity index is equal to ind (M ). For k = 1, it becomes the usual imprimitivity index of a matrix [12, chapter 8]. Thus, a family is k-primitive if and only if n = 1, i.e., the maximal partition is trivial. Remark 2 The existence and uniqueness of the maximal partition (Theorem 1 and Proposition 1), on which matrices A1 , . . . , Ak act as commuting permutations, generalize the corresponding results of the Perron-Frobenius theory characterizing non-primitive matrices [12, chapter 8]. The relation of the cardinality n of the maximal partition (the k-imprimitivity index) to the index of the lattice ind (M ) (Proposition 1) generalizes the Romanovsky theorem [20], according to which the imprimitivity index of a matrix is equal to the g.c.d. of the lengths of cycles of its digraph G. Indeed, in case k = 1 the lattice M is one-dimensional, and ind (M ) is the g.c.d. of its elements. Therefore, the equality n = ind (M ) in case k = 1 becomes the Romanovsky theorem. Since the k-imprimitivity index n does not exceed d, Proposition 1 implies that for every partition, the order of the corresponding group of permutations does not exceed d. This, however, does not mean that k must be less than or equal to d, because different matrices from A may define the same permutation. The simplest situation is when F is a group of cyclic permutations, in which case all permutations defined by matrices As are powers of one cyclic permutation of Q. The group F may also be a direct product of several cyclic permutations. This already exhausts all possible cases, because every finite abelian group is a direct product of cyclic groups with prime-power orders (see, for instance, [1]). Several illustrative examples are analyzed in the next section. Theorem 2 There is a polynomial-time algorithm for deciding k-primitivity. For an arbitrary nonnegative matrix family A = {A1 , . . . , Ak } satisfying (a) and (b), the algorithm determines the k-primitivity and finds maximal partition Ω = ∪ni=1 Ωi . The complexity ( the log2 d ) 3 of the algorithm is less than 2 k d 1 + 2d + 21 k 2 d 2 arithmetic operations. For sparse matrices, when( every column contains at most p nonzero elements, p < d, the complexity is ) k 2 less than k d 2p + log2 d + 2 . The algorithm and the proof of Theorem 2 are presented in Section V. Remark 3 The size of the instance of the algorithm is K = k d 2 , so its complexity is less than K 2 . With respect to the matrix size d, the complexity estimate is O(d 3 ). Let us remember that the fastest known algorithm of multiplication of two matrices takes already O(d 2.376 ) operations [5]. In practice, because of a large constant, that algorithm works slower than classical Strassen’s algorithm that takes O(d 2.807 ) operations [22]. 6

Remark 4 There is an interesting relation between the criterion of k-primitivity provided by Theorem 1 and recent results for a different type of primitivity obtained in [19]. A nonnegative matrix family A = {A1 , . . . , Ak } is called primitive if there is at least one strictly positive product (with repetitions permitted) of matrices from A. This concept provides a different generalization of primitivity of one matrix to matrix families. Clearly, primitivity implies k-primitivity, but not vice versa. For example, a pair of matrices A = {A1 , A2 } defined as follows:     0 0 1 0 1 0 A1 =  1 0 0  ; A2 =  1 0 0  0 1 0 0 0 1 is not primitive, because both A1 and A2 are permutation matrices, and hence, each their product, being a permutation matrix, cannot be strictly positive. On the other hand, it is checked directly that their Hurwitz product A (2,2) is strictly positive, and so A is k-primitive. The following theorem gives a sharp criterion of primitivity of nonnegative matrix families. Theorem B [19]. Suppose the family A is irreducible and its matrices have neither zero rows nor zero columns. Then A is not primitive if and only if there is a nontrivial partition r Ω = ∪i=1 Ωi of the set Ω = {1, . . . , d}, on which all matrices from A act as permutations. Thus, matrices of a non-primitive family also constitute permutations of a suitable partition. The difference is that those permutations do not have to commute. Moreover, even if a family A is neither primitive nor k-primitive, the corresponding partitions from Theorem 1 and Theorem B may be different. Besides, the assumptions of Theorem B is a bit stronger than condition (b): the matrices As have neither zero rows nor zero columns. In spite of similarity of Theorem B and Theorem 1, their proofs are totally different. The proof of Theorem B is based on geometrical properties of affine operators on polyhedra, while the proof of Theorem 1 is purely combinatorial.

III. Examples We consider two examples illustrating Theorem 1 and Proposition 1. Then we present Example 3 showing that without condition (b) both these assertions may fail. Example 1 The following family:  0 0 1  1 0 0 A1 =   0 1 0 0 0 1



 0 1   ; 0  0

A2

=

0  0   1 0

1 0 0 1

0 1 0 0

 0 0   1  0

is not k-primitive, because these matrices act as commuting permutations of the partition Ω1 = {1, 4}, Ω2 = {2}, Ω3 = {3}. The matrix A1 defines the cyclic permutation of these three sets, A2 is the inverse cyclic permutation. We have F ∼ = Z3 . 7

Note that after inserting a new colored arc to the corresponding digraph the family becomes k-primitive. Indeed, an extra positive entry leads to concatenation of two sets of the partition. We come to a new set Q′ and a new group of permutations F ′ . Since F ′ ⊂ F , it follows hat F ′ is a subgroup of F , in particular |F ′ | is a divisor of |F | = 3. Proposition 1 yields |F ′ | = |Q′ | ≤ 2, hence |Q′ | = 1, and therefore the new partition Q′ is trivial. Example 2 The family  A1

0  1   0 0

=

1 0 0 0

0 0 0 1

 0 0   ; 1  0

 A2

=

0  0   1 0

0 0 0 1

1 0 0 0

 0 1   0  0

is not k-primitive; the corresponding partition is Ωi = {i} , i = 1, 2, 3, 4. If we put numbers 1, . . . , 4 around the vertices of a rectangle, then the permutation A1 defines an axial symmetry of that rectangle, and A2 defines its central symmetry. Consequently, A1 and A2 generate the group of isometries of a rectangle, which is the Klein four-group V = Z2 ⊕ Z2 . Thus, F ∼ = V . Unlike Example 1, after inserting a new colored arc to the corresponding digraph the family remains not k-primitive. Indeed, an extra positive entry leads to concatenation of two sets of the partition, which is equivalent to identifying two vertices of the rectangle. Identifying now two other vertices, we obtain two points, on which V acts as Z2 . Thus, after inserting a new colored arc the family A still has a partition, this time into two sets. The following example shows the significance of condition (b). Example 3 A pair A = {A1 , A2 } of 6 × 6-matrices is defined by the corresponding twocolored digraph G. The arcs of the first color form the path 1 → 2 → 3 → 4 → 5; the arcs of the second color form the path 5 → 1 → 6 → 4. 22

33

11

44

66

55

The digraph G is strongly connected, whence A is irreducible. It has two cycles α = (1, 2, 3, 4, 5) and β = (1, 6, 4, 5), for which (α) = (4, 1)T and (β) = (1, 3)T . We have M = { x(4, 1)T + y(1, 3)T | x, y ∈ Z}. Since ind(M ) = 11, we see that M ̸= Z2 , and hence A is not k-primitive. Nevertheless, Theorem 1 fails for this family: there is no nontrivial partition of the set Ω = {1, . . . , 6}, on which A1 , A2 act as commuting permutations. We prove this by assuming the contrary. Let Ω = ∪ni=1 Ωi be such a partition, Q = {Ω1 , . . . , Ωn }, F be the abelian group generated by permutations A1 and A2 of the set Q. Since this group acts transitively on Q (because G is strongly connected), it follows that ( ) |F | = n.( A homomor) 2 2 T phism φ : Z → F is defined on the generators of Z as φ (1, 0) = A1 , φ (0, 1)T = A2 . 8

The kernel M ′ of φ is a lattice that contains M . Therefore, ind(M ′ ) is a divisor of ind(M ). Since ind(M ) = 11, we see that either ind(M ′ ) = 1 or ind(M ′ ) = 11. The former means that M ′ = Z2 , and the partition is trivial. Whence, ind(M ′ ) = 11, and M ′ = M . This yields that F is isomorphic to Z2 |M , and consequently, |F | = ind(M ) = 11. Thus, n = 11, which is impossible, because n ≤ 6. IV. Proofs of the main results In this section we give proofs of Theorem 1 and of Proposition 1. The proof of Theorem 2 and the algorithm for finding the maximal partition and determining the k-primitivity are presented in the next section. First, let us observe that it is sufficient to impose condition (b) only on columns. Indeed, if we prove Theorem 1 for matrices without zero columns, then it will imply the case of matrices without zero rows by merely taking transpose matrices and the inverse permutations. Thus, in the proofs of Theorem 1 and 2 and of Proposition 1 we assume that matrices do not have zero columns. This means that each vertex of the multigraph G has outgoing arcs of each of the k colors. The proof of Theorem 1 requires some short preliminary lemmas. For the moment, we consider an arbitrary nonnegative family A and impose only the assumption that it is irreducible (condition (a)). The following notation will be used: M is the lattice generated in Zk by color vectors of cycles of the multigraph G, γβ is a concatenation of paths γ and β on G, γ[a] is the terminal vertex of a path γ starting at the vertex a. If γ is of length zero, then γ[a] = a. For x, y ∈ Zk we write x ≡ y if x − y ∈ M . Consider the following equivalence relation on the set Ω. Elements a and b are equivalent (we write a ∼ b) if there is a path γ from a to b such that (γ) ∈ M . Lemma 1 The relation ∼ is an equivalence relation on Ω. Proof. We need to show that (i) a ∼ a, (ii) a ∼ b ⇒ b ∼ a (reflexivity), and (iii) a ∼ b , b ∼ c ⇒ a ∼ c (transitivity). (i). A cycle of length zero establish the equivalence a ∼ a. (ii). If a ∼ b, then there is a path γ from a to b such that (γ) ∈ M . By the irreducibility, there is a path β from b to a. Since γβ is a closed path, we have (γβ) = (γ) + (β) ∈ M . Hence (β) ∈ M , and so b ∼ a. (iii). If there are paths γ from a to b and β from b to c such that (γ), (β) ∈ M , then the path γβ goes from a to c and (γβ) = (γ) + (β) ∈ M . 2 The equivalence ∼ splits the set Ω into disjoint equivalence classes Ω1 , . . . , Ωn , where n ≥ 1. These classes were first introduced and analyzed in [9] (so-called imprimitivity classes) to characterize 2D strongly connected digraphs. We are going to see that, under additional assumption (b), these classes actually constitute the desired partition. To prove this we need first to establish several auxiliary facts formulated in Lemmas 2 - 4. Some of 9

them were observed earlier in [9], but we give their proofs for the sake of completeness. First, let us show that for every path γ connecting two elements of one class the corresponding color vector (γ) belongs to M . Lemma 2 If a ∼ b and b = γ[a], then (γ) ∈ M . Proof. Since b ∼ a, we have a = β[b] for some path β such that (β) ∈ M . Since γβ is a closed path, (γ) + (β) ∈ M , and hence (γ) ∈ M . 2 Lemma 3 Suppose a ∼ b, and a′ = γ[a], b′ = β[b]; then a′ ∼ b′ if and only if (γ) ≡ (β). Proof. We have b = δ[a] for some path δ such that (δ) ∈ M . By the irreducibility, there is a path γ ′ from a′ to a. Since γ γ ′ is a closed path, we have (γ) + (γ ′ ) ∈ M . Therefore, if (γ) ≡ (β), then (γ ′ ) + (δ) + (β) ≡ − (γ) + (δ) + (β) ≡ (δ) ≡ 0. Whence, (γ ′ δβ) ∈ M , and a′ ∼ b′ . Conversely, if a′ ∼ b′ , then by Lemma 2 we have (γ ′ δβ) ∈ M , and therefore 0 ≡ (γ ′ ) + (δ) + (β) ≡ (γ ′ ) + (β) ≡ −(γ) + (β), which completes the proof. 2 Lemma 3 means that for every elements a, b from one class Ωi and for every paths γ, β starting at a and b respectively, the relation (γ) ≡ (β) is equivalent to the fact that γ[a] and β[b] belong to the same class. Now let us show that the partition Q = {Ωi }ni=1 is trivial (i.e., n = 1) if and only if the family A is k-primitive. Lemma 4 Suppose all matrices of the family A are nonzero; then A is k-primitive if and only if n = 1. Proof. If n ≥ 2, then M ̸= Zk . Indeed, an arbitrary pair of elements from different classes are connected by a path (by irreducibility), whose color vector does not belong to M . Whence (Theorem A) the family A is not k-primitive. Conversely, if A is not k-primitive, then M ̸= Zk (Theorem A), and therefore M cannot contain all basic vectors ei , i = 1, . . . , k, of Zk (the ith entry of ei is one, all others are zeros), because they generate the whole Zk . Thus, ej ∈ / M for some j. Since the matrix Aj is nonzero, the multigraph G has an arc of jth color. Assume that it goes from some vertex a to b. Thus, b = γ[a] and (γ) ≡ ej ∈ / M. By Lemma 2, this implies a ̸∼ b, consequently, n ≥ 2. 2 Thus, if A is not k-primitive, then the partition is nontrivial. It remains to show that all matrices Ai define permutations of this partition, and these permutations commute with each other. We now invoke condition (b) in the proof. This was not necessary before: Lemmas 1 – 4 hold for every irreducible family of nonzero matrices, possibly, with some zero rows and columns. Proof of Theorem 1. By Lemma 4 the partition Ω = ∪ni=1 Ωi is nontrivial, i.e., n ≥ 2, provided A is not k-primitive. Let us first show that every matrix Am ∈ A defines a permutation of this partition. For a given set Ωi consider all arcs of the mth color starting at vertices a of G such that a ∈ Ωi . By Lemma 3 all these arcs lead to vertices that are equivalent 10

to each other, i.e., belong to one set Ωj of the partition. Consider the map fm : Q → Q defined as fm (Ωi ) = Ωj . By assumption (b) on columns, each vertex has an outgoing arc of mth color. Therefore, fm is well-defined (this is the only place in the proof, where we use assumption (b)). Let us show that fm is injective, in which case this is a permutation. If fm (Ωi1 ) = fm (Ωi2 ), then taking a ∈ Ωi1 , b ∈ Ωi2 , we have a ̸∼ b, and there are two equivalent paths γ, β (two arcs of color m) such that γ[a] ∼ β[b]. By irreducibility, there is a path γ ′ from γ[a] to a and a path β ′ from β[b] to b. Since γ γ ′ and β β ′ are both closed paths, we have (γ) + (γ ′ ) ∈ M and (β) + (β ′ ) ∈ M . Hence (γ ′ ) ≡ (β ′ ). However, the paths γ ′ and β ′ take two equivalent elements to not-equivalent ones, which contradicts Lemma 3. Thus, each matrix Am defines a permutation fm on the set Q. Finally, by Lemma 3, fm1 fm2 (Ωi ) = fm2 fm1 (Ωi ) for each m1 , m2 , i, hence all permutations fm commute, from which the theorem follows. 2 In the proof of Proposition 1 we use two facts from elementary group theory that follow from the group stabilizer theorem (see, for instance, [6]). If an abelian group of permutations F acts transitively on a finite set Q, then |F | = |Q|. Moreover, if f (a) = a for some f ∈ F, a ∈ Q, then f is the identity permutation. Proof of Proposition 1. Consider an arbitrary partition Q′ = {Ω′i } of the set Ω, on which matrices from A act as commuting permutations. Let F ′ be the corresponding abelian group generated by A. A map φ : Zk → F ′ is defined as φ(x) = Ac1 · · · Acm , where c is an arbitrary string such that (c) = x. By commutativity, this map is a well-defined homomorphism. Its kernel M ′ is a lattice in Zk that contains color vectors of all cycles of G. Indeed, if γ is a cycle and (γ) = z, then γ[a] = a for some a ∈ Ω. Suppose a ∈ Ω′s ; then for the string c corresponding to the path γ starting at a, the permutation φ(z) = Ac1 · · · Acm ∈ F ′ has a fixed point Ω′s . Since the group F ′ is abelian, it follows that φ(z) is the identity permutation, and so z ∈ M ′ . Therefore, M ⊂ M ′ , and hence F ′ is a subgroup of F , and |F ′ | is a divisor of |F |. If M = M ′ , then the the partition Q′ coincides with Q, because the elements of the partition are the equivalence classes defined by M (Lemma 1). In this case the kernel of φ is M , and hence F ′ = F = Zk |M . Otherwise, if M is a proper subset of M ′ , then F ′ is a proper subgroup of F , consequently |F ′ | < |F | and |F ′ | is a divisor of |F |. Since the group F is abelian, we have |F | = n. 2 V. The algorithm for deciding the k-primitivity and for finding the maximal partition To prove Theorem 2 we present the algorithm for finding the k-imprimitivity index n and the maximal partition. Then we complete the proof by justifying the algorithm and ¯ = by estimating its complexity. In this section we denote the maximal partition by Q ¯ ¯ {Ω1 , . . . , Ωn }. The instance of the algorithm is a family of nonnegative d × d-matrices A = {A1 , . . . , Ak } satisfying (a) and (b). The condition (b) is imposed on columns (i.e., the matrices As do 11

not have zero columns, see explanations in Section IV). Since everything depends on the location of the positive entries, we may restrict ourselves to 0 − 1 matrices. The output is ¯ = {Ω ¯ 1, . . . , Ω ¯ n } of the set Ω = {1, . . . , d} and a multigraph J with the maximal partition Q n vertices and with k colors of arcs. That multigraph describes the permutations of the ¯ defined by our matrices: there is an arc of the rth color from a vertex i to a vertex set Q ¯ i) = Ω ¯ j , where fr is the permutation defined by the matrix Ar . The j if and only if fr (Ω family A is k-primitive if and only if n = 1. Notation and the idea of the algorithm. The algorithm consists of three parts. We start with the partition of Ω to one-element sets: Q = {Ωi }di=1 , Ωi = {i} , i = 1, . . . , d. The first part. We take the first set Ω1 and the first matrix A1 ∈ A. We consider the full image A1 Ω1 which is the support of the first column of A1 . If we find two points a, b ∈ A1 Ω1 that belong to different sets of our partition, say a ∈ Ωi , b ∈ Ωj , i ̸= j, then we conclude ¯ We that the sets Ωi and Ωj must be contained in one set of the maximal partition Q. concatenate the sets Ωi and Ωj , i.e., we put Ωi = Ωi ⊔ Ωj and remove the set Ωj from our partition. Thus, the concatenation updates the partition as follows: it replaces two sets of the partition Ωi , Ωj by one set Ωi ⊔ Ωj , all other sets remain the same. After that we have a new partition, the number of its elements is less by 1. Now we restart the algorithm with the new partition. If the image A1 Ω1 is contained in one set of our current partition, then we do not change the partition and take the next set Ω2 , do the same with the image A1 Ω2 , etc. When all sets of the partition are exhausted, we take the second matrix A2 , etc. This way we run over all images A t Ωj for t = 1, . . . , k and Ωj ∈ Q. The image A t Ωj is the union of supports of all columns of A t , whose indices belong to Ωj . Every time, when we find two elements of the image that belong to two different sets of the partition Q, we update the partition Q by concatenation of these two sets, and then start the algorithm again by checking A1 Ωj for all Ωj ∈ Q. The first part terminates, if we checked the images A t Ωj for all pairs t = 1, . . . , k , Ωj ∈ Q, and did not update the partition. This means that we obtain a partition Q = {Ωj1 , . . . , Ωjr }, on which every matrix A t defines a map ft as follows: ft (Ωa ) = Ωb if At Ωa ⊂ Ωb . In the second part we check the injectivity of the maps ft . We run over all pairs (At , Ωj ) , A t ∈ A , Ωj ∈ Q. If we find a pair for which the map ft has at least two preimages of some set Ωj , i.e., ft (Ωa ) = Ωj and ft (Ωb ) = Ωj , then we concatenate the sets Ωa and Ωb . Then we restart the algorithm (going back to the first part) with the new partition. The second part terminates, if we exhaust all pairs (At , Ωj ) and did not make any concatenation. In this case all the maps ft are injective. i.e., they are permutations. Thus, after the second part all matrices of A act as permutations of Q, and we go to the third part. In the third part we check that all permutations ft commute. We run over all pairs (t1 , t2 ) such that 1 ≤ t1 < t2 ≤ k and over all sets Ωj ∈ Q. If for some t1 , t2 , Ωj we have ft1 ft2 (Ωj ) ̸= ft2 ft1 (Ωj ), then the sets Ωa = ft1 ft2 (Ωj ) and Ωb = ft2 ft1 (Ωj ) must be ¯ We concatenate Ωa and Ωb , and restart the contained in one set of the maximal partition Q. algorithm (going back to the first part) with the new partition. The third part terminates when ft1 ft2 (Ωj ) = ft2 ft1 (Ωj ) for all t1 , t2 , Ωj . After at most d − 1 concatenations, we come to a partition, for which all the three parts of the algorithm terminate. Then the algorithm terminates, the final partition Q coincides 12

¯ = {Ω ¯ 1, . . . , Ω ¯ n } up to the order. with the maximal partition Q Now we present the algorithm. The algorithm. Zero step. We have the set Ω = {1, . . . , d} and its partition into d elementary sets Ωi = {i}, i = 1, . . . , d. To this partition we associate a set of d pairs {(i, i) | i = 1, . . . , d}. (m) Main loop. The mth step, 1 ≤ m ≤ d − 1. We have a partition of the set Ω into d − m + 1 disjoint sets Ωj 1 , . . . , Ωj d−m+1 . We also have the corresponding set of d pairs {(i, j) | i ∈ Ωj , i = 1, . . . , d}. We consider a multigraph J with d − m + 1 vertices called 1, . . . , d − m + 1. Its arcs can be colored in k different colors numbered by 1, . . . , k. At the beginning the set of arcs is empty. We set t = 1, r = 1. (t) For t = 1, . . . , k (r) For r = 1, . . . , d − m + 1 We take all columns of the matrix At , whose indices belong to Ωj r and take the set of all indices {is } of positive entries of these columns (a number is belongs to this set if at least one of these columns has a positive element in the is th row). We take the second element j of the pair (i1 , j) and compare it with second elements of all other pairs (is , ·) , s ≥ 2. If we find a pair (is , q) with q ̸= j, then we conclude that the sets Ωj and Ωq must ¯ We concatenate the sets Ωj and Ωq , be contained in one set of the maximal partition Q. i.e., we put Ωj = Ωj ⊔ Ωq and remove Ωq from the partition. This means that in all pairs {(i, q) | i ∈ Ωq }, we replace the second index q by j. We thus obtain a partition into d − m sets; the set Ωq disappears being absorbed by Ωj . This concatenation also updates the graph J: we identify two vertices j and q to one vertex with index j (and add all the corresponding arcs) and remove the vertex q. Then the mth step is over. If m = d − 1, then we set m = d and go to the Final step (f ). Otherwise, if m < d − 1, then we go to the (m + 1)st step (m). Otherwise If all the second indices are equal to j, then If the vertex j of the multigraph J has an incoming arc h → j of color t, then Ωh ¯ Indeed, the matrix At and Ωj r must be contained in one set of the maximal partition Q. ¯ maps both Ωh and Ωj r to one set Ωj , and hence, to one set of the maximal partition Q. On the other hand, At acts injectively on the maximal partition, hence Ωh and Ωj r must be contained in one set of the maximal partition. We concatenate the sets Ωh and Ωj r . The mth step is over. If m = d − 1, then we set m = d and go to the Final step (f ). Otherwise, if m < d − 1, we go to the (m + 1)st step (m). Otherwise If the vertex j has no incoming arcs of color t, then we insert an arc j r → j of color t to the multigraph J. Let us recall that jr and j are indices such that At Ωjr ⊂ Ωj . End If End If End For (r) End For (t) 13

For all pairs of numbers t1 , t2 such that 1 ≤ t1 < t2 ≤ k. For j ∈ {j1 , . . . , j d−m+1 } We check whether or not At1 At2 Ωj = At2 At1 Ωj . To this end we consider a t1 t2 -path in the multigraph J from the vertex j and a t2 t1 -path from the same vertex j. Assume that they lead to vertices a and b respectively. If a ̸= b, then, by commutativity, the sets Ωa and Ωb must be contained in one set ¯ We concatenate the sets Ωa and Ωb . The mth step is over. If of the maximal partition Q. m = d − 1, then we set m = d and go to the Final step (f ). Otherwise, if m < d − 1, then we go to the (m + 1)st step (m). End If End For End For (f ) Final step If m ≤ d − 1, then the family A is not k-primitive, n = d − m + 1, and the n sets ¯ Ωj r , r = 1, . . . , n form the maximal partition Q. Otherwise If m = d, then n = 1, the maximal partition is trivial, and the family A is k-primitive. End If The end of the Algorithm Before we justify the Algorithm and give the proof of Theorem 2 let us consider the following illustrative example. Example 4 Let k = 2, d = 4,  0  1 A1 =   0 0

the family A = {A1 , A2 }  0 1 1 0 0 0   ; A2 = 1 0 0  1 0 0

be defined as follows:   0 1 0 0  1 0 0 0     0 0 1 1  0 0 0 0

The algorithm starts with the elementary partition Ωi = {i} , i = 1, 2, 3, 4. Set m = 1, take the matrix A1 (t = 1) and the set { Ω1 (r = 1). The first column } of A1 has only one positive entry (A1 )21 . Whence, the set (i, j) i ∈ Ω1 , (A1 )ij > 0 contains only one pair (1, 2). Obviously, there are no two pairs with two different second indices j, therefore we leave the partition as it is and consider the next set Ω2 = {2}. Now we have two pairs (2, 3) and (2, 4), because the matrix A1 contains two positive elements in the second column, their indices are 3 and 4. This means that the image A1 Ω2 intersects two sets of the partition: Ω3 and Ω4 . We concatenate Ω3 and Ω4 : we set Ω3 = Ω3 ∪ Ω4 = {3, 4} and remove Ω4 . Thus, we have a new partition: Ω1 = {1} , Ω2 = {2} , Ω3 = {3, 4}. We restart the Algorithm with this partition and set m = 2. Again we start with t = 1, r = 1, i.e., with the pair (A1 , Ω1 ). This time, for every sets Ω1 , Ω2 , Ω3 , the image A1 Ωi is contained in one set of the partition, and 14

we do not make any concatenations. Then we take t = 2, i.e., the matrix A2 , and similarly check the images A2 Ωi , i = 1, 2, 3. We have A2 Ω1 = Ω2 , A2 Ω2 = Ω1 , A2 Ω3 = {3} ⊂ Ω3 . We see that every image is contained in one set of the partition, hence, we keep the same partition Q = {Ω1 , Ω2 , Ω3 }. Now we conclude that each matrix At , t = 1, 2, defines a map ft : Q → Q and check the injectivity of those maps. The map f1 takes Ω1 , Ω2 , Ω3 to Ω2 , Ω3 , Ω1 respectively, hence it is injective. The map f2 takes Ω1 , Ω2 , Ω3 to Ω2 , Ω1 , Ω3 respectively, hence it is also injective. Thus, we do not make any concatenation and keep the same partition Q, on which both f1 , f2 are permutations. The last part: we check commutativity of those permutations. Since k = 2, we have only one pair (t1 , t2 ) = (1, 2). For every j = 1, 2, 3 we check the equality f1 f2 (Ωj ) = f2 f1 (Ωj ). It is violated already for j = 1: f1 f2 (Ω1 ) = Ω3 and f2 f1 (Ω1 ) = Ω1 . We concatenate the sets Ω1 and Ω3 and get a new partition: Ω1 = {1, 3, 4} and Ω2 = {2}. With this partition we restart the Algorithm going to the first step: set m = 3 and r = 1, t = 1. Looking over all pairs (r, t) , r = 1, 2 , t = 1, 2, we find a pair r = 1, t = 2 for which the image A2 Ω1 = {2, 3} intersects two different sets: Ω1 and Ω2 . We concatenate Ω1 and Ω2 and get a trivial partition: Ω1 = {1, 2, 3, 4}. The Algorithm terminates. The maximal partition is trivial, the imprimitivity index n = 1, and the pair A = {A1 , A2 } is k-primitive. Proof of Theorem 2. First we prove that the final partition constructed by the Algo¯ = {Ω ¯ 1, . . . , Ω ¯ n }. rithm coincides, up to the order of elements, with the maximal partition Q Then we derive the upper bound for the complexity of the Algorithm. Denote the final partition by Q = {Ωj 1 , . . . , Ωj p }. Consider the last step (m) of the Algorithm. Since at each step, except for the last one, the number of elements of the partition decreases by 1, the last step has number m = d−p+1. At that step the Algorithm does not make any concatenation, hence that step starts with the same partition Q = {Ωj 1 , . . . , Ωj p }. In the cycles (t) and (r) the Algorithm realizes two things: 1) for every t = 1, . . . , d and r = 1, . . . , p it checks that the image At Ωj r is contained in one set of the partition; otherwise it concatenates two sets, 2) it checks that the matrix At does not send elements from two different sets of the partition (Ωjr and some Ωh ) to one set, otherwise it concatenates Ωjr and Ωh . Since we do not make concatenations at the last step, it follows that each matrix At defines an injective map ft : Q → Q. This means that ft is a permutation. In the cycle “For all t1 , t2 such that 1 ≤ t1 < t2 ≤ k” the Algorithm checks that for every j ∈ {j1 , . . . , j p } we have ft1 ft2 Ωj = ft2 ft1 Ωj , otherwise it concatenates the sets ft1 ft2 Ωj and ft2 tt1 Ωj . Since the Algorithm does not make any concatenation at the last step, it follows that all permutations ft commute. Thus, the matrices A1 , . . . , Ak act as commuting permutations on the final partition Q. It remains to show that this partition ¯ To this end we prove that each set of the final partition Q is contained in coincides with Q. ¯ This will imply, by the maximality of Q, ¯ that Q ¯ = Q. some set of the maximal partition Q. Let us show by induction on m that the partition obtained after the (m − 1)st step of the Algorithm possesses this property: every its set is contained in some set of the maximal partition. For m = 1, i.e., for the initial partition Ωi = {i} , i = 1, . . . , d, this obviously obtained after the (m − 1)st step. At holds. Assume this holds for the partition {Ωjr }d−m+1 r=1 the mth step we concatenate two sets of this partition. This happens in one of the following 15

three cases: 1) If we find two elements i1 , is ∈ Ωj r that are sent by a matrix At to two different sets Ωj and Ωq respectively, then we concatenate Ωj and Ωq . Since, by induction, the sets Ωj r , Ωj , Ωq ¯jr, Ω ¯ j, Ω ¯ q respectively), it are contained in sets of the maximal partition (denote them by Ω ¯ j r , hence, At maps both i1 and is to one set of the maximal partition, follows that i1 , is ∈ Ω ¯ ¯ q . Thus, Ωj and Ωq are both contained in Ω ¯ j . Whence, after the and therefore, Ωj = Ω concatenation of Ωj and Ωq , the new partition still possesses this property. 2) If we find sets Ωj r , Ωh and Ωj such that At maps both Ωj r and Ωh to Ωj , then we concatenate Ωj r and Ωh . Again we conclude that Ωj r and Ωh must be contained in one set of the maximal partition, hence, after their concatenation the new partition keeps this property. 3) If we find matrices At1 , At2 ∈ A and a set Ωj such that At1 At2 Ωj = Ωa , At2 At1 Ωj = Ωb and Ωa ̸= Ωb , then we concatenate Ωa and Ωb . By the inductive assumption, the sets ¯ j, Ω ¯ a, Ω ¯ b respecΩj , Ωa , Ωb are contained in sets of the maximal partition, denote them by Ω ¯ j = At2 At1 Ω ¯ j , hence Ω ¯a = Ω ¯ b . Consequently, the sets tively. By the commutativity, At1 At2 Ω Ωa and Ωb are contained in one set of the maximal partition, and after their concatenation the new partition still possesses this property. Thus, we have proved that the final partition constructed by the Algorithm coincides with the maximal partition. To complete the proof of Theorem 2 we need to estimate the complexity of the algorithm. Assume every column of our matrices contains at most p ≤ d nonzero entries. For each i ∈ Ωj r , we take the ith column of matrix At and write down all pairs (i, j), where j is an index of the positive entry of this column. For a given i it takes at most p operations, and for all i ∈ Ωj r it takes takes at most p |Ωj r | operations. The same number of operations is needed for comparing the second elements of pairs for the set (i, j) , i ∈ Ωj r . For all matrices At ∈ A, we have in total 2kp |Ωj r | operations, and for all sets Ωjr , we have 2kp |Ω| = 2kpd. Since the algorithm performs at most d − 1 steps, we have 2kpd 2 operations. Furthermore, for each r = 1, . . . , d − m + 1 we have to decide whether the vertex j has an incoming arc of the color t. Since by this iteration the multigraph J has r − 1 arcs of the color t, this problem is reduced to deciding whether j belongs to a given set of r −1 numbers, which takes at most log2 r operations. Adding one arc to J is one operation. Thus, each iteration takes at most 1+log2 r = log2 2r operations. By∑ summing over all r = 1, . . . , d−m+ ∑d−m+1 1, t = 1, . . . , k and over all m = 1, . . . , d−1, we obtain k d−1 log2 2r ≤ k d 2 log2 d m=1 r=1 operations. To check commutativity we need to look over all triples (t1 , t2 , j), where 1 ≤ t1 < t2 ≤ d−1 and 1 ≤ j ≤ d − m + 1, and for each of them compare the corresponding numbers a and b. (d − m + 1) operations. By summing over all steps m = 1, . . . , d − 1, we This takes k(k−1) ] [ 2 k(k−1) d(d−1) get 2 − 1 ≤ 14 k 2 d 2 operations. 2 Finally, we make at most d − 1 concatenations. Each concatenations of two sets Ωi and Ωj takes at most max {|Ωi | , |Ωj |} ≤ d − 1 operations. Thus, all concatenations cost in total (d − 1)2 operations.

16

By summing we estimate the total complexity: 2kpd

2

1 + k d log2 d + k 2 d 2 + (d − 1)2 4 2

≤

kd

2

(

k) 2p + log2 d + . 2 2

Acknowledgements. The author is grateful to both anonymous Referees for their attentive reading and many valuable comments and suggestions.

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