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Exponents of primitive companion matrices. (English) Zbl 1423.15032
A square nonnegative (positive) matrix $$A$$, i.e., their entries are nonnegative (positive) real numbers, is called primitive if there exists a positive integer number $$m$$ such that $$A^{m}$$ is a positive matrix. The smallest among such positive integer numbers is said to be the exponent of $$A,$$ denoted by $$\exp (A).$$
For a class $$X$$ of nonnegative matrices of size $$n\times n$$, the exponent set $$E_{n}(X)= \{m\in N : \text{ there exists a primitive matrix } A\in X , \exp (A)=m\}.$$ A sharp upper bound for the exponent of a primitive matrix is $$\omega_{n}= (n-1)^{2} +1,$$ the so called Wielandt bound (for a proof, see [H. Schneider, Linear Algebra Appl. 353, No. 1–3, 5–10 (2002; Zbl 1006.15023)]). For a given class $$X$$ of non negative matrices, the problem of finding $$E_{n}(X)$$, bounds on $$E_{n}(X)$$ and matrices in $$X$$ such that the bounds are attained constitute a relevant challenge.
Let us denote by $$C_{n}$$ the set of all $$(0,1)$$ companion matrices of polynomials $$x^{n} - \sum_{k=0}^{n-1} a_{k} x^{k},$$ where $$a_{k}\in \{0,1\}.$$ The subset of all primitive $$(0,1)$$ companion matrices is denoted by $$CP_{n}.$$
In this paper the authors deduce the cardinal of $$CP_{n}$$ (Theorem 2.1) as well as the exponent of a matrix $$A\in CP_{n}$$ assuming that the trace of $$A$$ is positive (Theorem 2.4). In a next step, the exponents of matrices in $$CP_{n}$$ with zero trace are obtained (see Theorem 3.2). Indeed, if $$A\in CP_{n},$$ then $$n\leq \exp (A) \leq \omega_{n}.$$ Moreover, there exists a matrix $$B\in CP_{n}$$ such that $$\exp (B)=n.$$ Finally, some elements of $$E_{n}( CP_{n})$$ are described.
The techniques of directed graph (digraph) theory are extensively used in the proofs. Let us remind that for a (0,1)-matrix $$A= [a_{i, j}]_{i, j=1}^{n}$$ of size $$n\times n,$$ the adjacency digraph $$D(V ,E)$$ has as vertex set $$V=\{1,2, \cdots, n\}$$ and as edge set $$E= \{(i, j): a_{i, j}=1\}.$$ On the other hand, for a digraph $$D$$ its adjacency matrix $$A$$ has as entries $$a_{i,j}=1$$ if there is an edge from $$i$$ to $$j$$ in $$D$$ and $$a_{i, j}=0$$ otherwise.
##### MSC:
 15B36 Matrices of integers 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
##### Keywords:
primitive matrix; companion matrix; digraph
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