Matrices with totally signed powers

A square real matrix A is said to have signed d-power, if the sign pattern of the power Ad 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. A is said to be d-powerful if all the non-zero terms in the expansion formula of each entry of Ad have the same sign. A is powerful if A is d-powerful for all positive integers d. We show that A has totally signed powers is equivalent to A being powerful, although A has signed d-power is not equivalent to A being d-powerful.