Bases of primitive nonpowerful sign patterns (Q443725)

The sign pattern of a real square matrix is the associated \((0,\pm1)\)-matrix whose entries are the signs of the entries of the given matrix. When calculating the square (or higher powers) of a sign pattern, indeterminate signs arise when adding a plus and a minus; these are given the symbol \(\#\). Clearly the number of sign patterns of a given real square matrix is finite, so for a given sign pattern there exist positive integers \(l\), \(p\) such that \(A^l = A^{l + p}\) is the first repeated pair in the sequence \(A,A^2,A^3, \dots\). The integer \(l\) is called the base of \(A\) and \(p\) is called the period of \(A\). Furthermore, \(A\) is said to be powerful if none of its powers contains a \(\#\), otherwise nonpowerful. Finally, it is said to be primitive if the matrix obtained from \(A\) by changing each of its minuses to a plus has the property: some power consists of nothing but 1's. The authors show that the base \(l\) of a primitive nonpowerful sign pattern \(A\) is the least positive integer \(l\) such that every entry of \(A^i\) is \(\#\). The proof uses graph theoretical methods, which are often useful when studying powers of square matrices.NEWLINENEWLINEThe associated digraph \(D(A)\) of a sign pattern \(A\) of order \(n\) has vertex set \(V= \{1,2,\dots,n\}\) and edge set \(E= \{(i,j) \mid a_{ij} \neq 0\}\). The associated signed digraph \(S(A)\) of \(A\) is obtained from \(D(A)\) by directing each edge according to the sign of \(a_{ij}\). The authors study the connections between \(A\) and \(S(A)\). For a primitive nonpowerful sign pattern \(A\) of order \(n\) and with base \(l > \frac{3}{2} n^2 - 2n+4\), some properties of the cycles in \(A(A)\) are obtained and some upper bounds are derived. In the last section, sign patterns with special bases are discussed.