# Hermitian and positive definite matrices

Suppose A is a Hermitian invertible matrix with positive diagonal entries. When A will become a positive definite or its all the eigen values will be positive?

-

The statement is not true: take $A:=\pmatrix{1&a\\a&1}$, where $a>1$: the eigenvalues are $1+a$ and $1-a<0$, and $A$ is invertible.
If $A$ is strictly diagonally dominant, $i.e.$ if $|a_{ii}| > \sum_{i \neq j} |a_{ij}|$ for all $i$, and all of the diagonal entries of $A$ are positive then $A$ will be positive definite.
However, there can be matrices which are positive definite and are not strictly diagonally dominant. In general, $A$ will be positive definite iff all of its eigenvalues are positive.