positive definite

Positive definite

A structure is positive definite when it comes with a positive-definite functional $F:X\to [0,\infty )$, such that:

$$F(x)=0⟺x=0$$

… zero only at the distinguished zero (or–for groups–identity) element, and is positive elsewhere.
Examples:
norms: $\Vert x\Vert =0⟺x=0$ (if it can fail, → seminorm)
metrics: $d(x,y)=0⟺x=y$ (if it can fail → pseudometric)
inner products: $⟨x,x⟩>0⟺x=0$ (if allowing zeros→ positive semidefinite)

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.

Link to original

https://en.wikipedia.org/wiki/Positive_definiteness
https://mathworld.wolfram.com/PositiveDefiniteMatrix.html