Rank(AB) = Rank(A)Rank(B)?The Rank Product Theorem for Matrices

[itpro]

Can someone point me to the proof or give it here for the claim that product of the two positive definite (real) matrices is positive definite.

How about determinants of two matrices? Is det(AB) = det(BA)

Rank(AB) = Rank(A)Rank(B)?

Thank you in advance.

This is not a homework question. If you need a proof or a source for my claim please send PM.

 

[Karamata]

Can someone point me to the proof or give it here for the claim that product of the two positive definite (real) matrices is positive definite.

http://en.wikipedia.org/wiki/Matrix_multiplication#Hadamard_product

The Hadamard product of two positive-(semi)definite matrices is positive-(semi)definite

How about determinants of two matrices? Is det(AB) = det(BA)

You can use: https://www.physicsforums.com/showthread.php?t=94344