Positive Definite Matrices

A symmetric matrix $A$ is positive definite if:

$$\mathbf{x}^TA\mathbf{x} > 0 \quad \text{for all } \mathbf{x} \neq \mathbf{0}$$

Equivalent Conditions

All of these are equivalent:

1. All eigenvalues $> 0$
2. All pivots $> 0$
3. All upper-left determinants $> 0$ (Sylvester’s criterion)
4. $\mathbf{x}^TA\mathbf{x} > 0$ for all $\mathbf{x} \neq 0$

Tests (In Order of Speed)

1. Pivots: all pivots positive? (Fastest)
2. Determinants: all leading minors positive?
3. Eigenvalues: all positive?
4. Energy: $\mathbf{x}^TA\mathbf{x} > 0$ for every $\mathbf{x}$?

Applications

• Minimum: $A$ positive definite → $f(x) = \frac{1}{2}\mathbf{x}^TA\mathbf{x} - \mathbf{b}^T\mathbf{x}$ has a unique minimum
• Covariance matrices are positive semi-definite
• Optimization: second derivative test