Positive Definite Matrices

Key Takeaways

• Positive definite: $\vec{x}^T A \vec{x} > 0$ for all $\vec{x} \neq \vec{0}$
• Equivalent to all eigenvalues being positive
• Cholesky decomposition $A = LL^T$ is 2× faster than LU
• Covariance matrices and kernel matrices must be PSD
• Hessian must be PD at a local minimum for optimization