Positive definite

Dependencies:

1. Matrix

Let $A$ be a real $n$ by $n$ matrix.

Then $A$ is positive definite (PD) iff $$\mathrm{\forall }u\in {\mathbb{R}}^n-\{0\},u^{T}Au>0$$ Then $A$ is positive semidefinite (PSD) iff $$\mathrm{\forall }u\in {\mathbb{R}}^n,u^{T}Au\ge 0$$ Then $A$ is negative definite (ND) iff $$\mathrm{\forall }u\in {\mathbb{R}}^n-\{0\},u^{T}Au<0$$ Then $A$ is negative semidefinite (NSD) iff $$\mathrm{\forall }u\in {\mathbb{R}}^n,u^{T}Au\le 0$$

Dependency for:

1. Sum of positive definite matrices is positive definite
2. Positive definite iff eigenvalues are positive

Info:

• Depth: 3
• Number of transitive dependencies: 4

Transitive dependencies:

1. Group
2. Ring
3. Semiring
4. Matrix