Positive definite

Dependencies:

1. Matrix

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

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

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