Real Symmetric Matrices

Definition: Real Symmetric Matrix

A real symmetric matrix is a symmetric matrix $A\in {\mathbb{R}}^{n\times n}$ over the real numbers.

Definition: Definiteness of a Real Symmetric Matrices

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

Theorem: Definiteness Criteria

A real symmetric matrix $M\in {\mathbb{R}}^{n\times n}$ whose eigenvalues are all real is:

• positive definite if and only if all of its eigenvalues are positive;

• positive semi-definite if and only if all of its eigenvalues are non-negative;

• negative definite if and only if all of its eigenvalues are negative;

• negative semi-definite if and only if all of its eigenvalues are non-positive;

• indefinite if and only if it has both positive and negative eigenvalues.

PROOF

TODO