TITLE: Sufficient conditions for positive semidefinite matrices
QUESTION [5 upvotes]: Is a symmetric matrix with positive terms (i.e., $a_{ij} > 0$) and positive determinant positive semidefinite?
Is a symmetric matrix with positive terms, positive determinant, and
terms that satisfy $a_{ij}^2 \leq a_{ii} a_{jj}$ positive semidefinite?

REPLY [3 votes]: b) The following statements are equivalent:

The symmetric matrix $A$ is positive semidefinite
All eigenvalues are non negative
All the principal minors of $A$ are non negative. 
There exists a $B$ such that $A = B^{T}B$ where $B$ can be thought of as a square root of a positive operator.

[1] Kreyszig, Erwin. Introductory functional analysis with applications. Vol. 1. New York: wiley, 1989.