Combinatorial structure of $k$-semiprimitive matrix families

The notion of a primitive matrix plays an important role in the Perron-Frobenius Theory of nonnegative matrices and in the closely related theory of Markov chains. We recall that a nonnegative matrix is said to be primitive if some power of this matrix contains only positive entries. There are several generalizations of the property of being primitive for matrix families. For example, in [1], a family of nonnegative matrices of the same order is said to be primitive if the multiplicative semigroup generated by this family contains a positive matrix. A more general definition of primitivity is related to so-called Hurwitz products (see [2], for example).

Definition 1.  Let a family $\mathscr A=\{A_1,\dots,A_k\}$ of nonnegative matrices of the same order and a family of nonnegative integers $\alpha=\{\alpha_1, \dots,\alpha_k\}$ be given. By the Hurwitz product corresponding to $\alpha$ we mean the matrix $\mathscr A^\alpha$ equal to the sum of all the products of matrices in $\mathscr A$ that contain precisely $\alpha_q$ occurrences of the matrix $A_q$, $q=1,\dots,k$.

For example, if $\mathscr A=\{A_1,A_2,A_3\}$, then $\mathscr A^{(2,0,1)}=A_1^2A_3+A_1A_3A_1+A_3A_1^2$.

Definition 2.  A family of nonnegative $(n\times n)$-matrices $\mathscr A= \{A_1,\dots, A_k\}$ is said to be $k$-primitive if $\mathscr A^\alpha$ is positive for some family $\alpha=\{\alpha_1, \dots,\alpha_k\}$.

The property of $k$-primitivity was introduced in [3]; for a bibliography of subsequent works on this topic, see [4].

Corresponding to the family $\mathscr A$ in a natural way is the coloured directed multigraph with vertex set $N=\{1,\dots,n\}$ whose arcs are coloured using colours in the set $\{1,\dots,k\}$. Here an arc $ij$ of colour $q$ exists if $(A_q)_{ij}>0$. A family of numbers $\alpha=\{\alpha_1,\dots,\alpha_k\}$ is referred to as a colour vector or a colouring of a path if an arc of $q$th colour occurs in this path $\alpha_q$ times, $q=1,\dots,k$. For short, by a colouring we mean an arbitrary tuple of nonnegative integers whose length is equal to the number of matrices in the family. A coloured graph is said to be $k$-primitive if there is a colouring $\alpha$ such that each vertex of the graph is accessible from every vertex by a path of colouring $\alpha$. It can readily be seen that $\mathscr A$ is $k$-primitive if and only if the coloured graph of $\mathscr A$ is $k$-primitive.

For a nonnegative matrix $A$ of order $n$ and a subset $L\subseteq N$ we set $A(L)={\{j\in N\mid(A)_{ij}>0,\,i\in L\}}$. A family $\mathscr A$ is said to be reducible if there is a proper subset $L\subset N$ such that $A(L)\subseteq L$ for every matrix $A\in\mathscr A$. A family is irreducible if and only if the corresponding coloured graph is strongly connected. It is clear that $k$-primitivity is a special case of irreducibility.

Now we present an algebraic criterion for $k$-primitivity; this criterion was given in [3] in the case of $k=2$ and in [5] and [6] for an arbitrary $k$. Let $\sigma_1,\dots,\sigma_r$ be all simple contours (that is, closed paths without repeating vertices, except for the first and the last) in the coloured graph of $\mathscr A$. We denote the colourings of these contours by $(\sigma_1),\dots,(\sigma_r)$. We consider the integral sublattice of the lattice $\mathbb Z_k$ generated by the colourings of the simple contours, that is, the set

$$\begin{equation*} M=\biggl\{\sum_{l=1}^r w_l(\sigma_l)\Bigm| w_l\in\mathbb Z,\, l=1,\dots,r\biggr\}. \end{equation*}$$

Theorem A.  (see [3], [5] and [6]) An irreducible family $\mathscr A$ of nonzero matrices is $k$-primitive if and only if $M=\mathbb Z_k$.

In [4] a new approach to the problem of $k$-primitivity was suggested; this approach helped to generalize to matrix families the results in Perron-Frobenius Theory concerning the structure of an irreducible imprimitive matrix.

Theorem B.  (see [4]) Let the irreducible family $\mathscr A$ satisfy at least one of the following conditions: the matrices in it contain no zero rows or the matrices in it contain no zero columns. Then $\mathscr A$ is not $k$-primitive if and only if there exists a nontrivial partition of the set $N$ on whose classes matrices in the family act by mutually commuting permutations.

Let us clarify the assertion of Theorem B under the assumption that the matrices in the family contain no zero rows but can have zero columns. For a matrix $A$ of order $n$ suppose that the partition of $N$ has the following property: corresponding to each class $L$ of the partition is another class $M$ such that $A(L)\subseteq M$. As a result, $A$ defines a map of the set of classes of the partition. If this map is bijective, then, in the terminology of [4], the matrix $A$ acts on the partition as a permutation. In [4] a maximal partition was described (that is, one with the largest number of classes possible) such that matrices in a family satisfying the assumptions of the theorem act by commuting permutations of the classes of that partition. This means that using a single permutation similarity all matrices in the family can be reduced to a block form so that each block row and each block column contains a unique nonzero block. Furthermore, the positions of nonzero blocks in the matrices $AB$ and $BA$ coincide for any $A$ and $B$ in the family. If the family consists of a single imprimitive (that is, not primitive) matrix $A$, then the block form described in [4] coincides with the one defined by the Perron-Frobenius Theorem (see [7], for example).

Both the formulation of Theorem B and its proof are of combinatorial nature. Thus the line of the theorem of Romanovsky (see [8], and also [9]), who discovered the combinatorial nature of the imprimitivity index, extends to a more general situation. Using Theorem B, Protasov [4] constructed an effective algorithm for the recognition of $k$-primitivity which could not be designed by following old approaches.

The domain of application of the notion of $k$-primitivity contains, in particular, the multidimensional Markov chains, where this notion plays a role similar to that of primitive matrices in the classical theory of Markov chains (see [10]). Apart from primitive stochastic matrices, the theory of Markov chains also considers regular stochastic matrices. The significance of this class of matrices is as follows. Let ${P=(p_{ij})}$ be an $(n\times n)$ stochastic matrix (with respect to rows), that is, a nonnegative matrix with the property $\sum_{j=1}^np_{ij}=1$, $i=1,\dots,n$. In the theory of Markov chains, the entry $p_{ij}$ is treated as the probability of the transition from the state $i$ to the state $j$ in one step. The probabilities of transitions in $k$ steps are described by the matrix $P^k=(p^{(k)}_{ij})$. A chain is said to be ergodic if the limit $\lim_{k\to\infty} p_{ij}^{(k)}=\pi_j$ exists and is independent of $i$ for all $i$ and $j$. For the ergodicity of a chain it is necessary and sufficient that for some exponent $k$ the matrix $P^k$ contain a positive column. A stochastic matrix with this property is said to be regular (see [11], [9] and [7]). Thus, primitivity is a special case of regularity. In this and only this special case, all the limits $\pi_j$ are positive.

It should be noted that for stochastic matrices the notion of regularity is often identified with the notion of primitivity (see, for example, [12]). To avoid any ambiguity of terminology, we refer to a nonnegative (not necessarily stochastic) matrix some power of which contains a positive column as a semiprimitive matrix.

The following definition plays the main role in the present paper.

Definition 3.  A family of nonnegative matrices $\mathscr A= \{A_1,\dots,A_k\}$ is said to be $k$-semiprimitive if the matrix $\mathscr A^\alpha$ contains a positive column for some family $\alpha=\{\alpha_1,\dots,\alpha_k\}$.

While $k$-primitivity generalizes the classical notion of primitive matrix, $k$-semiprimitivity is a generalization of the semiprimitivity property of a nonnegative matrix.

A coloured graph is said to be $k$-semiprimitive if there exists a colouring $\alpha$ such that some vertex of the graph is accessible from any other vertex by a path of colouring $\alpha$. It is clear that a matrix family is $k$-semiprimitive if and only if the coloured graph of the family is $k$-semiprimitive.

Suppose in what follows that the following condition holds:

a) matrices in the families under consideration have no zero rows.

For a $k$-coloured graph this means that arcs of $k$ distinct colours issue from every vertex. This implies the following lemma.

Lemma 1.  In a coloured graph a path with any prescribed colouring goes out of each vertex.

We say that a vertex $j$ is $\alpha$-accessible from a vertex $i$ if there exists a path of colouring $\alpha$ which goes from $i$ to $j$. Let us introduce several further definitions.

Vertices $i$ and $j$ are

• 1)  
  $\alpha$-compatible at a vertex $l$ if $l$ is $\alpha$-accessible from these vertices;
• 2)  
  $\alpha$-compatible if some common vertex is $\alpha$-accessible from these vertices;
• 3)  
  colour compatible at a vertex $l$ if there are paths of the same colouring from the vertices $i$ and $j$ to $l$;
• 4)  
  colour compatible if there are paths of the same colouring from $i$ and $j$ to the same vertex.

Proposition 1.  A coloured graph is $k$-semiprimitive if and only if every two vertices of the graph are colour compatible.

Proof.  The necessity of the condition follows immediately from the definition of $k$-semiprimitivity. Let us prove sufficiency. Assume that the vertices 1 and 2 are compatible at a vertex $l$ by paths of colouring $\alpha_1$, and let a path of colouring $\alpha_1$ (which exists by Lemma 1) go from the vertex 3 to a vertex $m$. The vertices $l$ and $m$ are compatible by paths of some colouring $\alpha_2$. Thus, there are paths of colouring $\alpha_1+\alpha_2$ from the vertices $1$, $2$ and $3$ to some common vertex. Continuing in this way, we see that some vertex of the graph is accessible from any other vertex by a path of colouring $\alpha_1+\alpha_2+\dots+\alpha_{n-1}$.

Remark.  In the above proof (and below) we use the (naturally defined) operation of concatenation (or joining) for paths and the fact that the colourings are added under the concatenation of paths. Moreover, the following property is used: if vertices are $\alpha$-compatible, then they are $\beta$-compatible for any $\beta\geq\alpha$ (the inequality is treated componentwise).

A vertex of a graph is called an $\alpha$-focus if it is $\alpha$-accessible from any vertex. A vertex is referred to as a focus if it is an $\alpha$-focus for some $\alpha$. We note two simple properties of foci.

Property 1.  A vertex accessible from a focus is also a focus.

Property 2.  If a vertex is an $\alpha$-focus then it is a $\beta$-focus for every $\beta\geq\alpha$.

Consider the relationships between the $k$-primitivity and $k$-semiprimitivity of a coloured graph.

Proposition 2.  For the $k$-primitivity of a coloured graph it is necessary and sufficient that the graph be $k$-semiprimitive and strongly connected. In other words, a matrix family is $k$-primitive if and only if it is $k$-semiprimitive and irreducible.

Proof.  The `if' part of the assertion follows immediately from the definition of $k$-primitivity. The converse assertion follows from the properties 1 and 2. It follows from Property 1 that all vertices of a strongly connected $k$-semiprimitive graph are foci. Let the $i$th vertex be an $\alpha(i)$-focus. If $\beta\geq\alpha(i)$ for $i=1,2,\dots,n$, then by Property 2 all vertices are $\beta$-foci. That is, the $k$-primitivity property holds for the colouring $\beta$.

We recall that every subset of vertices of a graph generates the subgraph whose arcs are the ones joining vertices in this subset between themselves. Below we consider only subgraphs generated by subsets of vertices. A subset of vertices (and the subgraph generated by it) is said to be closed if vertices not belonging to this subset are not accessible from vertices in it. A subset is closed if and only if each arc beginning in it ends in it as well.

Proposition 3.  A coloured graph is $k$-semiprimitive if and only if it contains a unique closed $k$-primitive subgraph whose vertices are accessible from every vertex of the graph.

Proof.  Let a graph be $k$-semiprimitive. It follows readily from the definition of focus and from Property 1 that the subgraph generated by the foci is closed, $k$-primitive (and hence strongly closed), and its vertices are accessible from every vertex of the graph. To prove uniqueness we consider an arbitrary closed strongly connected subgraph. This subgraph contains all foci of the graph because otherwise some foci would not be accessible from some vertices of the subgraph, which contradicts the definition of a focus. In fact, the subgraph under consideration coincides with the subgraph generated by the foci because a strongly connected graph cannot contain proper closed subgraphs.

Let us prove the converse assertion. By assumption, there is a path from any vertex of the graph to the set of vertices of the primitive subgraph. Every extension of this path is contained in the same set by the closedness property. Hence we can choose paths so that a path from each vertex $i$ to some vertex $f(i)$ of the primitive subgraph has the same colouring for all $i$, say, a colouring $\alpha$. As the subgraph is primitive, there are paths of some (common) colouring $\beta$ from the $f(i)$ to an arbitrary vertex $j$ of this subgraph. It is now clear that we can arrive at the vertex $j$ following a path of colouring $\alpha+\beta$ from every vertex of the graph.

Example 1.  Let $\mathscr A$ be a $k$-primitive family and let $\mathscr B$ be an arbitrary irreducible family containing $l$ matrices. We form a reducible family of all possible matrices of the form

$$\begin{equation} \begin{pmatrix} A &0 \\ X &B \end{pmatrix},\qquad A\in\mathscr A,\quad B\in\mathscr B, \end{equation} \tag{ 2.1 }$$

where $X\ne 0$ for at least one of the matrices. It can readily be seen that the coloured graph of the family (2.1) satisfies the assumptions of Proposition 3, and hence this family is $m$-semiprimitive, $m=kl$.

If the family $\mathscr A$ is $k$-semiprimitive, then by Proposition 1 every two vertices of the coloured graph of the family are colour compatible. In the general case, the notion of colour compatibility defines a binary relation on the vertex set $N$. Obviously, this relation is reflexive and symmetric for every coloured graph. One can readily give an example of a coloured graph with nontransitive colour compatibility relation. Let us introduce a type of matrix families for which the colour compatibility relation is transitive and thus an equivalence relation.

Definition 4.  We say that a vertex $i$ of a coloured graph is entire if every two vertices accessible from $i$ by paths of the same colouring are colour compatible. A coloured graph is said to be entire if all its vertices are entire. A matrix family is entire if the coloured graph of the family is entire.

The condition that a family be entire is rather weak. In particular, we shall show that all irreducible families of matrices satisfy it.

We denote the colour compatibility relation by the symbol $\mathscr C$. For brevity, we shall write `$(i,j)$-path' when speaking of a path from the vertex $i$ to the vertex $j$.

Lemma 2.  The binary relation $\mathscr C$ on the vertices of an entire coloured graph is an equivalence.

Proof.  It suffices to prove transitivity. Let $i\,\mathscr Cj$ and $j\,\mathscr Cl$. Then an $(i,p)$-path and a $(j,p)$-path of some colouring $\alpha$ lead to some vertex $p$. There also exist a $(j,q)$-path and an $(l,q)$-path of some colouring $\beta$. We claim that $i\,\mathscr Cl$. Let us extend the $(j,p)$-path by an arbitrary $(p,u)$-path of colouring $\beta$ and extend the $(j,q)$-path by an arbitrary $(q,v)$-path of colouring $\alpha$. We have now two paths of colouring $\alpha+\beta$ which begin at $j$. As the vertex $j$ is entire, $u$ and $v$ are colour compatible, that is, there are paths from $u$ and $v$ to some vertex $w$ which have the same colouring $\gamma$. We can finally certify that there are paths from $i$ and $l$ to $w$ which have colouring $\alpha+\beta+\gamma$; hence $i\,\mathscr Cl$.

Lemma 3.  Each coloured graph contains some entire vertices.

Proof.  Suppose that some coloured graph has no entire vertices. Then it has vertices that are not colour compatible. Let

$$\begin{equation} i_1,i_2,\dots,i_m \end{equation} \tag{ 3.1 }$$

be a maximal family of pairwise colour-incompatible vertices. There are paths of some colouring $\alpha$ from a nonentire vertex $i_1$ to some colour incompatible vertices $j_0$ and $j_1$. Let $j_2,\dots,j_m$ be vertices which are $\alpha$-accessible from $i_2,\dots,i_m$, respectively. The vertices

$$\begin{equation} j_0,j_1,j_2,\dots,j_m \end{equation} \tag{ 3.2 }$$

must be pairwise colour-incompatible. Indeed, for example, let the vertices $j_1$ and $j_2$ be $\beta$-compatible. Then $i_1$ and $i_2$ are $(\alpha+\beta)$-compatible, which is impossible because the list (3.1) contains no compatible vertices. We see that the vertices in (3.2) are pairwise incompatible, which contradicts the condition that the list (3.1) is maximal.

Lemma 4.  A vertex accessible from an entire vertex is also entire.

Proof.  Let a vertex $j$ be $\alpha$-accessible from an entire vertex $i$ and let vertices $p$ and $r$ be accessible from $j$ by paths of colouring $\beta$. Then $p$ and $r$ are colour compatible as vertices which are $(\alpha+\beta)$-accessible from the entire vertex $i$. This proves that $j$ is entire.

It follows from Lemmata 3 and 4 that all vertices in a strongly connected coloured graph are entire, that is, the following assertion holds.

Proposition 4.  Every strongly connected coloured graph is entire. In other words, every irreducible system of nonnegative matrices is entire.

The partition into equivalence classes of the relation $\mathscr C$ has the property described in Theorem B; the matrices of the entire family act on this partition as commuting permutations. To prove this fact, we need a lemma.

Lemma 5.  Let there be an $(i,p)$-path and a $(j,q)$-path of the same colouring $\alpha$. In this case,

1) if $p\,\mathscr Cq$, then $i\,\mathscr Cj$;

2) if $i\,\mathscr Cj$, then $p\,\mathscr Cq$.

Proof.  1) Let the vertices $p$ and $q$ be compatible by paths of colouring $\beta$. Then the vertices $i$ and $j$ are obviously compatible by paths of colouring $\alpha+\beta$.

2) Suppose that the vertices $i$ and $j$ are compatible at a vertex $r$ by paths of colouring $\beta$. Let us extend the $(i,r)$-path and the $(j,r)$-path by the same path of colouring $\alpha$ with the end at some vertex $v$. Let us extend an $(i,p)$-path and a $(j,q)$-path by arbitrary paths of colouring $\beta$; let the extended paths end at vertices $u$ and $w$, respectively. The vertices $u$ and $v$ are colour compatible because they are accessible from an entire vertex $i$ by paths of colouring $\alpha+\beta$, and the vertices $v$ and $w$ are colour compatible because they are accessible from an entire vertex $j$ by paths of the colouring $\alpha+\beta$. By the transitivity property of the relation $\mathscr C$ we have $u\mathscr Cw$. It follows from what was proved in 1) that this yields $p\,\mathscr Cq$.

Theorem 1.  The matrices of an entire family $\mathscr A$ act on the $\mathscr C$-classes as commuting permutations.

Proof.  Let us consider the coloured graph of the family $\mathscr A$. For paths of length 1, assertion 2) of Lemma 5 means that for every $\mathscr C$-class $L$ and every colour $q$ there is a unique $\mathscr C$-class $M$ containing the ends of all arcs of colour $q$ that begin in $L$. Thus, the colour $q$ (that is, the matrix $A_q\in\mathscr A$) defines a map on the set of $\mathscr C$-classes. For a path of an arbitrary length $k$, the sequence of colours of the arcs $q_1q_2\dotsb q_k$ (the product of the matrices $A_{q_1}A_{q_2}\dotsb A_{q_k}$) defines a map (on the set of $\mathscr C$-classes) depending on the colouring of the path only. This means that the actions of matrices in the family $\mathscr A$ on the set of $\mathscr C$-classes commute.

By part 1) of Lemma 5, if two parts of some colouring $\alpha$ end at the same $\mathscr C$-class, then both the initial vertices of these paths also belong to a common $\mathscr C$-class. Hence, the map of $\mathscr C$-classes defined by an arbitrarily taken colouring $\alpha$ is a permutation.

As was proved in [4] (Proposition 1), for an irreducible family of matrices there exists a unique partition of the set $N$ with the largest number of classes possible on which matrices in the family act by commuting permutations. (This partition was said to be maximal.) A similar assertion holds for a wider type of entire families of matrices.

Theorem 2.  Let $\mathscr A$ be an entire family of matrices. Then the partition into $\mathscr C$-classes is a subpartition of every partition of the set $N$ of vertices of the coloured graph of the family on which the matrices in $\mathscr A$ act as commuting permutations.

Proof.  Let us consider an arbitrary partition $\chi$ on which the matrices of the family act as commuting permutations. If the vertices $i$ and $j$ belong to distinct classes of $\chi$, then they are not colour compatible (that is, belong to distinct $\mathscr C$-classes), because any two paths of the same colouring define the same permutation of the classes of the partition $\chi$, and this takes the classes $[i]$ and $[j]$ to distinct classes. Hence, the partition into the $\mathscr C$-classes is a subpartition of $\chi$.

In other words, Theorem 2 claims that for an entire family of matrices the partition of the set $N$ into $\mathscr C$-classes is a unique partition into the largest number of classes possible on which matrices in the family act by commuting permutations. If a $k$-semiprimitive family $\mathscr A$ is irreducible, then it is $k$-primitive by Proposition 2, and the partition into $\mathscr C$-classes coincides with the maximal partition of [4]. In this case, the assertion of Theorem 1 is contained in Theorem B. To show that Theorem 1 generalizes Theorem B, we shall construct examples of entire reducible families. To this end, we need the following partial converse of Theorem 1.

Corollary 1.  Suppose that a partition $\chi$ of vertices of the coloured graph of the family $\mathscr A$ has the following properties:

1) the matrices in $\mathscr A$ act on $\chi$ by commuting permutations;

2) vertices in the same class of the partition $\chi$ are colour compatible.

Then $\mathscr A$ is entire, and $\chi$ coincides with the partition into $\mathscr C$- classes.

Proof.  By property 1) any two vertices which are accessible by paths of the same colouring from some vertex of the graph belong to the same class of the partition $\chi$. By property 2) these two vertices are colour compatible. Hence the family $\mathscr A$ is entire. It follows from property 1) and Theorem 2 that vertices belonging to distinct classes of $\chi$ are colour incompatible. By property 2) vertices belonging to the same class of $\chi$ are colour compatible. This proves the coincidence of the partition $\chi$ and the partition into $\mathscr C$-classes.

Example 2.  Let the family $\mathscr A$ consist of the matrices

$$\begin{equation*} A_1=\begin{pmatrix} 1 &0 &0 &0 &0 \\ 1 &1 &0 &0 &0 \\ 0 &0 &1 &0 &0 \\ 0 &0 &0 &1 &0 \\ 0 &0 &0 &0 &1 \end{pmatrix}\quad\text{and}\quad A_2=\begin{pmatrix} 0 &0 &1 &0 &0 \\ 0 &0 &0 &1 &0 \\ 1 &0 &0 &0 &0 \\ 0 &1 &0 &0 &0 \\ 0 &0 &0 &0 &1 \end{pmatrix}. \end{equation*}$$

Then the following properties of $\mathscr A$ can readily be verified. The coloured graph of the family is not strongly connected (it contains three components of strong connectedness that are generated by the sets of vertices $\{1,3\}$, $\{2,4\}$, and $\{5\}$). The matrices $A_1$ and $A_2$ act on the partition $C_1=\{1,2\}$, $C_2=\{3,4\}$, $C_3=\{5\}$ by commuting permutations. For the colouring $\alpha=(1,2)$ the matrix $\mathscr A^\alpha$ has the block diagonal form

$$\begin{equation*} \mathscr A^{(1,2)}=\begin{pmatrix} 2 &0 &0 &0 &0 \\ 1 &2 &0 &0 &0 \\ 0 &0 &2 &0 &0 \\ 0 &0 &1 &2 &0 \\ 0 &0 &0 &0 &2 \end{pmatrix}. \end{equation*}$$

Using this representation for this matrix we can conclude that the vertices belonging to the same class of the above partition are compatible by paths of colouring $(1,2)$.

Thus, $\mathscr A$ is reducible and, by Corollary 1, an entire family with colour compatibility classes $C_1$, $C_2$ and $C_3$.

We treat the inequality $A\leq B$ for nonnegative matrices of the same order elementwise. We say that block matrices are of the same type if they are decomposed into blocks in the same way and, moreover, the nonzero blocks in these matrices are similarly positioned.

Example 3.  Let $\mathscr A=\{A_1,\dots,A_k\}$ be a $k$-semiprimitive family of matrices of order $n$. We introduce the family $\mathscr B$ of matrices of order $2n$ of the following form:

$$\begin{equation*} B_q=\begin{pmatrix} 0 &A_q \\ E &0 \end{pmatrix},\qquad q=1,\dots,k. \end{equation*}$$

Obviously, each matrix $B_q$ acts by the same cyclic permutation of the classes $C_1=\{1,\dots,n\}$ and $C_2=\{n+1,\dots,2n\}$. Thus, the condition 1) in Corollary 1 holds for the partition $C_1$, $C_2$. We claim that condition 2) also holds. Let the colouring $\alpha=(\alpha_1, \dots,\alpha_k)$ be given. By the definition of the Hurwitz product, the matrix $\mathscr B^{2\alpha}$ is the sum of all possible products of matrices in $\mathscr B$ such that $B_q$ occurs $2\alpha_q$ times in each of these products. In particular, this sum contains products such that the matrix $\displaystyle{B_q^2=\begin{pmatrix} A_q &0 \\ 0 & A_q \end{pmatrix}}$ occurs $\alpha_q$ times in the product, $q=1,\dots,k$. Obviously, the sum of all these products is equal to $\smash{\displaystyle{\begin{pmatrix} A^\alpha &0 \\ 0 &A^\alpha \end{pmatrix}}}$. This implies that

$$\begin{equation} \begin{pmatrix} \mathscr A^\alpha &0 \\ 0 &\mathscr A^\alpha \end{pmatrix} \leq\mathscr B^{2\alpha}, \end{equation} \tag{ 3.3 }$$

and both sides of (3.3) are matrices of the same type. For a colouring $\alpha$ suppose that the Hurwitz product $\mathscr A^\alpha$ contains a positive column. It follows from (3.3) that every two vertices of the same class $C_i$, $i=1,2$, in the coloured graph of $\mathscr B$ are compatible by paths of colouring $2\alpha$; thus, condition 2) in Corollary 1 holds. This implies that $\mathscr B$ is entire, and $C_1$ and $C_2$ are classes of the relation $\mathscr C$.

The family $\mathscr B$ is irreducible if and only if $\mathscr A$ is. Indeed, it is known that a finite family of matrices is irreducible if the sum of the matrices in the family is an irreducible matrix. Let $A$ be the sum of the matrices in $\mathscr A$; then $\displaystyle{B=\begin{pmatrix} 0 &A \\ E &0 \end{pmatrix}}$ is the sum of the matrices in $\mathscr{B}$. The proof of the fact that the matrices $A$ and $B$ are reducible or irreducible simultaneously is simple, and we omit it. Thus, if a family $\mathscr A$ is reducible, then $\mathscr B$ is an example of an entire reducible non-$k$-semiprimitive family.

Suppose that an entire family $\mathscr A$ is not $k$-semiprimitive, that is, there is no Hurwitz product $\mathscr A^\alpha$ containing a positive column. We claim that in this case one can reduce all matrices in the family, by a single permutation similarity, to a form for which there exists a block-diagonal Hurwitz matrix, all of whose diagonal blocks contain a positive column. Note that $\mathscr A^{(1,2)}$ in Example 2 is such a matrix.

Let the entire family $\mathscr A$ and its coloured graph be given. Let $C_1,C_2,\dots,C_r$ be the set of $\mathscr C$-classes of vertices, which contain $n_1,n_2,\dots,n_r$ elements, respectively. We make a similarity transformation of matrices of $\mathscr A$ as follows: move the rows with the indices in $C_1$ to the first $n_1$ places, the rows with the indices in $C_2$ to the next $n_2$ places, and so on. We also transpose the columns in the same way. We decompose every transformed matrix of the family into blocks so that the diagonal blocks have orders $n_1,n_2,\dots,n_r$. Then we obtain a block-monomial matrix, that is, a matrix with precisely one nonzero block in every block row and in every block column. Namely, in the matrix corresponding to the $q$th colour, in the $u$th block row, the nonzero block lies in the $v$th column if the colour in question (that is, the matrix $A_q$) takes the class $C_u$ to the class $C_v$. It follows from condition a) (see §2) that the nonzero blocks of the monomial matrices contain no zero rows. We refer to the block form thus obtained for matrices in $\mathscr A$ as the $\mathscr C$-form. A reduction of matrices in $\mathscr A$ to the $\mathscr C$-form corresponds to re-indexing the vertices of the coloured graph of the family in accordance with their grouping into $\mathscr C$-classes.

In what follows we assume that the matrices in the entire family $\mathscr A$ have the $\mathscr C$-form. It is clear that matrices define the same permutation of $\mathscr C$-classes if and only if they have the same type. It follows from the proof of Theorem 2 that matrices corresponding to paths of colouring $\alpha$ have the same type. The Hurwitz product $\mathscr A^\alpha$ is a sum of matrices of the same type. Hence the following lemma holds.

Lemma 6.  The matrix $\mathscr A^\alpha$ is a block-monomial matrix of the same type as any product of matrices in $\mathscr A$ corresponding to a path of colouring $\alpha$.

The number $(A_q)_{ij}$ is called the weight of an arc $ij$ of colour $q$ in the coloured graph of $\mathscr A$, and by the weight of an $(i,j)$-path of colouring $\alpha$ we will mean the product of the weights of the arcs in this path. Then the $(i,j)$th entry of the matrix $\mathscr A^\alpha$ is equal to the sum of the weights of all possible $(i,j)$-paths of colouring $\alpha$.

Lemma 7.  Let $\mathscr A$ be an entire family of matrices, let $\alpha(1),\dots,\alpha(t)$ be an arbitrary family of colourings, and let $\alpha=\alpha(1)+\dotsb+\alpha(t)$. Then

1) $\mathscr A^{\alpha(1)}\dotsb\mathscr A^{\alpha(t)} \leq\mathscr A^\alpha$;

2) the matrices $\mathscr A^{\alpha(1)}\dotsb\mathscr A^{\alpha(t)}$ and $\mathscr A^\alpha$ are of the same type.

Proof.  Indeed, an $(i,j)$th element of the matrix $\mathscr A^\alpha$ is equal to the sum of the weights of all $(i,j)$-paths of colouring $\alpha$, while the $(i,j)$th element of $\mathscr A^{\alpha(1)}\dotsb\mathscr A^{\alpha(t)}$ is equal to the sum of the weights of the $(i,j)$-paths of colouring $\alpha$ of special form: namely, each of these paths is the concatenation of $t$ paths of colourings $\alpha(1),\dots,\alpha(t)$. This implies the inequality in 1).

To prove assertion 2) of the lemma we note that both the matrices in statement 2) are block-monomial. The first is block-monomial as a product of block-monomial matrices and the other is so by Lemma 6. Furthermore, the matrices satisfy the inequality in 1). It can readily be seen that this is possible for block-monomial matrices of the same type only.

The following assertion is a special case of Lemma 7.

Corollary 2.  Let $\mathscr A$ be an entire family of matrices and let $\alpha$ be an arbitrary colouring. Then

$$\begin{equation} (\mathscr A^\alpha)^t\leq\mathscr A^{t\alpha} \end{equation} \tag{ 3.4 }$$

for every positive integer $t$, and the matrices $(\mathscr A^\alpha)^t$ and $\mathscr A^{t\alpha}$ have the same type.

Let the entire family of matrices and the corresponding coloured graph be given. By an $\alpha$-focus of a $\mathscr C$-class we mean a vertex which is $\alpha$-accessible from each vertex in the class. A focus of a class is a vertex which is an $\alpha$-focus for some colouring $\alpha$. A focus of a class is said to be proper if it belongs to the class.

Theorem 3.  In the coloured graph of an entire non-$k$-semiprimitive family $\mathscr A$ each $\mathscr C$-class contains a proper focus. Moreover, there exists a colouring $\alpha$ such that each $\mathscr C$-class has a proper $\alpha$-focus, so that the nonzero blocks of the matrix $\mathscr A^\alpha$ lie on the diagonal, and each of them contains a positive column.

Proof.  The existence of a focus for every $\mathscr C$-class is proved by an argument similar to that used in the sufficiency part of Proposition 1. Further, each vertex that is accessible from a focus is obviously a focus of the same class. Therefore, if a class $C_u$ admits a $\sigma(u)$-focus, then for $\sigma\geq\sigma(u)$, $u=1,\dots,r$, there exists a $\sigma$-focus for each $\mathscr C$-class. This means that every nonzero block of the matrix $\mathscr A^\sigma$ contains a positive column. Since $\mathscr A^\sigma$ is block-monomial, there exists an exponent $t$ such that $(\mathscr A^\sigma)^t$ is block-diagonal, and the diagonal blocks contain positive columns. By Corollary 2, the same properties hold for $\mathscr A^{t\sigma}$. Thus, every $\mathscr C$-class has a proper $\alpha$-focus for $\alpha=t\sigma$.

We proceed with irreducible families of matrices. By the Perron-Frobenius Theorem, a nonnegative irreducible matrix is either primitive or can be reduced by some permutation of rows and the same permutation of columns to a block-monomial form

$$\begin{equation*} A=\begin{pmatrix} 0 &A_{12} &\dots &0 \\ \dots \\ 0 &0 &\dots &A_{r-1,r} \\ A_{r1} &0 &\dots &0 \end{pmatrix}, \end{equation*}$$

and the diagonal blocks in the matrix $A^r=\operatorname{diag}(A_{11}^{(r)}, A_{22}^{(r)},\dots,A_{rr}^{(r)})$ are primitive. Hence all the diagonal blocks in some power $A^r$ are positive. In other words, the multiplicative semigroup $\langle A\rangle$ generated by $A$ contains a block-diagonal matrix with positive diagonal blocks.

The classical notion of primitivity admits various generalizations to families of matrices. In [1] a family $\mathscr A$ of nonnegative matrices was said to be primitive if the semigroup $\langle\mathscr A\rangle$ generated by $\mathscr A$ contains a positive matrix. Every primitive family $\mathscr A=\{A_1,\dots,A_k\}$ is $k$-primitive, but the converse assertion fails to hold (see [4], Remark 4).

In [1] a generalization of the Frobenius Theorem was proved; it can be formulated as follows. Assume that the matrices in an irreducible family $\mathscr A$ contain no zero rows or columns. If $\mathscr A$ is not primitive, then the matrices in the family can be transformed by a common permutation similarity to a block-monomial form with the following property: the semigroup generated by the transformed family contains a block-diagonal matrix with positive diagonal blocks. The main theorem in [1] has a combinatorial formulation; however, its proof uses the geometry of convex polyhedra. For this reason, the authors of [1] posed the problem of finding a combinatorial proof. This proof was given in [13]. It is rather long, as also is the original proof. A short (less than a page) analytic proof of the main theorem has been presented quite recently in [14].

In the spirit of results of [1], one can prove the following result concerning the property of $k$-primitivity.

Theorem 4.  Let $\mathscr A$ be an irreducible non-$k$-primitive family. Then there exists a colouring $\beta$ such that in the coloured graph of the family each vertex in each $\mathscr C$-class is $\beta$-accessible from any other vertex in the same class. In other words, the matrix $\mathscr A^\beta$ is a block-diagonal matrix with positive diagonal blocks.

Proof.  An irreducible non-$k$-primitive family is entire and not $k$-semiprimitive. Therefore, Theorem 3 can be applied. Let $\mathscr A$ be equipped with a colouring $\alpha$ with the property described in Theorem 3. We denote by $F_\alpha$ the set of proper $\alpha$-foci of a class $C_u$. Let $i$ and $j$ be proper foci, but $i\in F_\alpha$ and $j\in C_u\setminus F_\alpha$. Since the graph is strongly connected, there exists an $(i,j)$-path, say, of colouring $\sigma$. Thus, the vertex $j$ is a proper $(\alpha+\sigma)$-focus of the class $C_u$.

By Lemma 5, if there is some $(i,j)$-path of colouring $\sigma$ ($i,j\in C_u)$, then all the paths of colouring $\sigma$ beginning in $C_u$ lead to $C_u$. Therefore, the proper $\alpha$-foci of the class $C_u$ are also proper $(\alpha+\sigma)$-foci of this class. Hence, we have a strict inclusion $F_\alpha\subset F_{\alpha+\sigma}$. Thus, we can extend the set of proper foci of the class $C_u$ which are accessible by paths of the same colouring until this set coincides with $C_u$ for some colouring $\delta$. This means that the diagonal block corresponding to the class $C_u$ in the matrix $\mathscr A^\delta$ is positive. The matrix $(\mathscr A^\delta)^t$ has the same property for every positive integer $t$. Since the matrix $\mathscr A^\delta$ is block-monomial, the exponent $t$ can be chosen so that the matrix $(\mathscr A^\delta)^t$ is block-diagonal. Then by Corollary 2 the matrix $\mathscr A^{t\delta}$ is also block-diagonal, that is, defines the identity map of the set of $\mathscr C$-classes. Thus, for each class $C_u$ there exists a colouring $\beta(u)$ such that

1) every proper focus of $C_u$ is $\beta(u)$-accessible from all vertices in this class;

2) the matrix $\mathscr A^{\beta(u)}$ defines the identity map of the $\mathscr C$-classes.

These properties mean that the matrix $\mathscr A^{\beta(u)}$ has block-diagonal form, and the block corresponding to the class $C_u$ is positive. Multiplying the matrices $\mathscr A^{\beta(u)}$ in an arbitrary fixed order and taking into account Lemma 7, we obtain the following inequality for two block-diagonal matrices of the same type:

$$\begin{equation} A^{\beta(u_1)}\dotsb A^{\beta(u_{r-1})}A^{\beta(u_r)} \leq\mathscr A^\beta, \end{equation} \tag{ 4.1 }$$

where $\beta=\sum_{u=1}^r \beta(u)$. The matrix on the left-hand side of (4.1) has a positive diagonal block in the $r$th position because this block is a product of blocks without zero rows, the last factor in which is positive. By (4.1) the diagonal block in the $r$th position in the matrix $\mathscr A^\beta$ is positive. However, the matrix $A^{\beta(u_r)}$ is chosen arbitrarily among the matrices of the same type as $\mathscr A^{\beta(u)}$, and hence all diagonal blocks in $\mathscr A^\beta$ are positive.

Let us illustrate Theorem 4 by the following simple example.

Example 4.  Let the family $\mathscr A$ be formed by the matrices

$$\begin{equation*} A_1=\begin{pmatrix} 0 &0 &0 &1 \\ 0 &0 &0 &1 \\ 1 &0 &0 &0 \\ 0 &0 &1 &0 \end{pmatrix}\quad\text{and}\quad A_2=\begin{pmatrix} 0 &0 &1 &0 \\ 0 &0 &1 &0 \\ 0 &0 &0 &1 \\ 0 &1 &0 &0 \end{pmatrix}. \end{equation*}$$

It can readily be seen that $\mathscr A$ is irreducible and that the matrices in $\mathscr A$ act on the partition $C_1=\{1,2\}$, $C_2=\{3\}$, $C_3=\{4\}$ by commuting permutations. By Corollary 1, the vertices belonging to distinct classes are not colour compatible. From the form of the matrix

$$\begin{equation*} \mathscr A^{(1,1)}=\begin{pmatrix} 1 &1 &0 &0 \\ 1 &1 &0 &0 \\ 0 &0 &2 &0 \\ 0 &0 &0 &2 \end{pmatrix} \end{equation*}$$

we can see that vertices in the class $C_1$ of the coloured graph of the system are $(1,1)$-compatible and, moreover, each of these vertices is $(1,1)$-accessible from any other vertex in this class.