The critical exponent for generalized doubly nonnegative matrices

It is known that the critical exponent (CE) for conventional, continuous powers of n-by-n doubly nonnegative (DN) matrices is . Here, we consider the larger class of diagonalizable, entrywise nonnegative n-by-n matrices with nonnegative eigenvalues (generalized doubly nonnegative (GDN)). We show that, again, a CE exists and is able to bind with a low-coefficient quadratic. However, the CE is larger than in the DN case; in particular, 2 for . There seems to be a connection with the index of primitivity, and a number of other observations are made and questions raised. It is shown that there is no CE for continuous Hadamard powers of GDN matrices, despite it also being for DN matrices.