Definite matrix

Definition

A symmetric matrix $A$ is said to be

• Positive definite $⟺$ $X^{T}AX>0$ for $X\ne 0$
• Non-negative definite $⟺$ $X^{T}AX\ge 0$ for $X\ne 0$ where $X^{T}AX$ is the Quadratic form

Checking with Eigendecomposition

$$A=\sum {\lambda }_i{e}_i{e}_i^T$$ $$X^{T}AX=\sum {\lambda }_i{x}^T{e}_i{e}_i^{T}x$$ $$=\sum {\lambda }_i{y}_i{y}_i^T$$ $$=\sum {\lambda }_i{y}_i^2>0⟺\forall {\lambda }_i\ge 0$$

Thus, if all the eigenvalues of a symmetric matrix if greater than the zero, then it is a positive definite matrix.