Positive definite matrix

A positive definite matrix $$A \in \mathbb{R}^{n\times n}$$ is whose quadratic form $$x^T A x$$ takes strictly positive values for all $$x\in\mathbb{R}^n$$, i.e., $$x^T A x > 0$$.

A matrix is called positive semidefinite if $$x^T A x \ge 0$$ for all $$x \in \mathbb{R}^n$$.