Definiteness of Real Symmetric Matrices

Definition: Definiteness of a Real Symmetric Matrix

A real symmetric matrix $M\in {\mathbb{R}}^{n\times n}$ is

• positive-definite if ${\vec{v}}^{\mathsf{T}}M\vec{v}>0$ for every vector $\vec{v}\in {\mathbb{R}}^n\setminus \{\vec{0}\}$;
• positive semi-definite if ${\vec{v}}^{\mathsf{T}}M\vec{v}\ge 0$ for every vector $\vec{v}\in {\mathbb{R}}^n\setminus \{\vec{0}\}$;
• negative-definite if ${\vec{v}}^{\mathsf{T}}M\vec{v}<0$ for every vector $\vec{v}\in {\mathbb{R}}^n\setminus \{\vec{0}\}$;
• negative semi-definit if ${\vec{v}}^{\mathsf{T}}M\vec{v}\le 0$ for every vector $\vec{v}\in {\mathbb{R}}^n\setminus \{\vec{0}\}$;
• indefinite if there are vectors $\vec{u},\vec{v}\in {\mathbb{R}}^n$ such that ${\vec{u}}^{\mathsf{T}}M\vec{u}>0$ and ${\vec{v}}^{\mathsf{T}}M\vec{v}<0$.