Result: Matrices with totally signed powers: Matrices with totally signed powers.

Similarly to the sign of a real number, the sign pattern of a real matrix \(A\), denoted by \(\text{sign}(A)\), is the \((0, 1, -1)\)-matrix obtained from \(A\) by replacing each entry by its sign. The set of real matrices with the same sign pattern as \(A\) is called the qualitative class of \(A\), and is denoted by \(Q(A)\). In this paper, the authors first introduce the notion of signed \(d\)-power. More precisely, a square real matrix \(A\) is said to have signed \(d\)-power, if the sign pattern of the power \(A^d\) is uniquely determined by the sign pattern of \(A\). \(A\) is said to have totally signed powers if \(A\) has signed \(d\)-powers for all positive integers \(d\). There is a related notion of powerful matrices which was first introduced and studied by \textit{Z. Li}, \textit{F. Hall} and \textit{C. Eschenbach} [Linear Algebra Appl. 212--213, 101--120 (1994; Zbl 0821.15017)]. \(A\) is said to be \(d\)-powerful, if all the nonzero terms in the expansion formula of each entry of \(A^d\) have the same sign. \(A\) is powerful if \(A\) is \(d\)-powerful for all positive integers \(d\). With these definitions, the main purpose of the paper is to study the relationship between the matrices with totally signed powers and powerful matrices. The authors see that this relationship is different from the relationship with signed \(d\)-power and \(d\)-powerful matrices. More precisely, they prove that \(A\) having totally signed powers is equivalent to \(A\) being powerful, although \(A\) having signed \(d\)-power is not equivalent to \(A\) being \(d\)-powerful.