The critical exponent for continuous conventional powers of doubly nonnegative matrices

We prove that there exists an exponent beyond which all continuous conventional powers of n-by-n doubly nonnegative matrices are doubly nonnegative. We show that this critical exponent cannot be less than n-2 and we conjecture that it is always n-2 (as it is with Hadamard powering). We prove this conjecture when n<6 and in certain other special cases. We establish a quadratic bound for the critical exponent in general.