real symmetric matrix

Positive Definite, Negative Definite, Indefinite

A real symmetric matrix $G\in {\mathbb{R}}^{n\times n}$ is positive definite if for all vectors $\mathbf{x}\in {\mathbb{R}}^n\setminus \{0\}$

$${\mathbf{x}}^T\cdot G>0$$

Similarily, if for all vectors $\mathbf{x}\in {\mathbb{R}}^n$

$${\mathbf{x}}^T\cdot G<0$$

then the matrix is called negative definite.
If the matrix is neither positive or negative, it’s called an indefinite matrix

Example: Diagonal matrices with positive entries on the diagonal are positve definite and vice versa.