Example of non-equivalence of the two PSD definitions

1. Is the product of two PSD matrices also PSD?
2. What is positive definite matrix with example?
3. Is the product of two symmetric positive definite matrices positive definite?
4. What is the difference between positive definite and positive semidefinite?

Is the product of two PSD matrices also PSD?

The product of two symmetric PSD matrices is PSD, iff the product is also symmetric. More generally, if A and B are PSD, AB is PSD iff AB is normal, ie, (AB)TAB=AB(AB)T.

What is positive definite matrix with example?

A square matrix is called positive definite if it is symmetric and all its eigenvalues λ are positive, that is λ > 0. Because these matrices are symmetric, the principal axes theorem plays a central role in the theory. If A is positive definite, then it is invertible and det A > 0.

Is the product of two symmetric positive definite matrices positive definite?

The product of two positive definite matrices is not necessarily positive definite; in fact, the product may NOT be Hermitian, and thus cannot be positive definite.

What is the difference between positive definite and positive semidefinite?

Definitions. Q and A are called positive semidefinite if Q(x) ≥ 0 for all x. They are called positive definite if Q(x) > 0 for all x = 0. So positive semidefinite means that there are no minuses in the signature, while positive definite means that there are n pluses, where n is the dimension of the space.