positive semidefinite

For a real symmetric matrix \(M\), it is positive definite if for all vectors \(x\), \(x^T M x > 0\). \(M\) is positive semi-definite if for all \(x\), \(x^T M x \geq 0\).

Note that the property \(x^T M x \geq 0\) does not require symmetry (see stack overflow)

If a matrix \(A\) is positive definite then the eigenvalues are all positive. Because for an eigenvector \(x\), \(0 < x^TAx = x^T \lambda x = \lambda ||x||_2^2\). And \(0 < ||x||_2^2\). So \(0 < \lambda\).