Positive Semidefinite

Synonyms

Positive definite

Definition

A symmetric m × m matrix K satisfying ∀ x ∈ c^m : x^∗Kx ≥ 0 is called positive semidefinite. If the equality only holds for $x =\vec{ 0}$ the matrix is positive definite.

A function k : X × X → c, X ≠ Ø, is positive (semi-) definite if for all m ∈ n and all x₁, …, x _m ∈ X the m × m matrix $\vec{K}$ with elements K _ij : = k(x _i , x _j ) is positive (semi-) definite.

Sometimes the term strictly positive definite is used instead of positive definite, and positive definite refers then to positive semidefiniteness.