Definite Matrix

Definition (Positive Definite Matrix)

A matrix $M\in {\mathbb{R}}^{n\times n}$ is called positive definite iff $\forall \vec{x}\in {\mathbb{R}}^n,{\vec{x}}^{T}M\vec{x}>0$. We denote this property by $M\succ 0$.

Definition (Positive Semi-definite Matrix)

A matrix $M\in {\mathbb{R}}^{n\times n}$ is called positive semi-definite iff $\forall \vec{x}\in {\mathbb{R}}^n,{\vec{x}}^{T}M\vec{x}\ge 0$. We denote this property as $M⪰0$.

Connection to Bilinear Forms

A symmetric bilinear form $B:V\times V\to \mathbb{R}$ is positive definite if its corresponding quadratic form $q(u)=B(u)(u)$ satisfies $q(u)>0$ for all $u\ne 0$. In a given basis, this is equivalent to its matrix representation being positive definite.