Critical Exponents: Old and New

Let P be a class of matrices, and let A be an m-by-n matrix in the class. The critical exponent of P, if it exists, with respect to some notion of continuous powering is the lowest power g(P) such that for any matrix B in P, B^t is in P for all t > g(P). This paper considers two questions for several classes P (including doubly nonnegative and totally positive): 1) does a critical exponent g(P) exist? and 2) if so, what is it? For those where no exact result has been determined, lower and upper bounds are provided