There seems to be a difference in definitions of "positive-definite" between pure and applied math literature. For instance:
Linear algebra textbook (Horn and Johnson, 1990):
x^*Ax>0 for all non-zero x
x^* refers to conjugate transpose
Machine Learning textbook (Bishop, 2005)
x'Ax>0 for all non-zero x (real domain is implied)
The main difference in the definitions is that when restricted to real domain, the first definition also requires that A is symmetric, but the last two don't.