Cholesky: A = LLᵀ, where L is lower triangular. lᵢᵢ = √(aᵢᵢ − Σₖ lᵢₖ²). lᵢⱼ = (aᵢⱼ − Σₖ lᵢₖlⱼₖ)/lⱼⱼ. Only works for symmetric positive definite (all eigenvalues > 0). Half the cost and twice the stability of LU.
How to Use
Enter an SPD matrix:
- A: Symmetric positive definite
- Output: L factor
- Verify: LLᵀ = A
SPD Check
A matrix is SPD if: (1) A = Aᵀ (symmetric), (2) xᵀAx > 0 for all x≠0. Equivalently: all eigenvalues positive. All leading principal minors positive. Cholesky exists ⟺ A is SPD.
Applications
- Least squares: AᵀAx = Aᵀb
- Covariance matrices
- Random variable generation
- Kalman filter
Step-by-Step Instructions
- 1Enter SPD matrix.
- 2View L factor.
- 3Verify LLᵀ = A.
- 4Check positive definiteness.
- 5Use for solving.