How can I show that the Hadamard product of REAL matrices is positive? By Hadamard product, I mean the componentwise product of positive real matrices. By positive, 
I mean that it is positive semidefinite. 
I am currently trying to show this but am not sure if it is possible to do it through defining operators and then their respective matrices. 
Is it possible to do this using operators? Thanks!

What you ask for is called the schur product theorem. A proof is contained in the following wikipedia article:
https://en.wikipedia.org/wiki/Schur_product_theorem
