Positive definite matrix if eigenvalue has positive and negative solutions

Question

If a matrix $A$ is Hermitian and its eigenvalues have positive and negative solutions, is it still considered to be positive definite? For example, is the following matrix positive definite?

Answer 1

No. All of the eigenvalues of a Hermitian matrix must be positive for the matrix to be positive definite. In your example, the eigenvalues of the matrix are $+1/2$ and $-1/2$, so the matrix is indefinite.

Answer 2

It is not positive definite. All eigen values of a psd (positive semi-definite) matrix are necessarily nonnegative. So your (with eigen values $\frac 1 2$ and $-\frac 1 2$) matrix is not even psd.