Cholesky Decomposition Calculator
Decompose symmetric positive definite matrices into L·L^T with step-by-step solutions
Matrix Input
Matrix Properties
Cholesky Decomposition Results
Lower Triangular Matrix L:
[5.0000, 0.0000] [3.0000, 3.0000]
Verification (L·L^T):
[25.0000, 15.0000] [15.0000, 18.0000]
This should equal the original matrix A
Step-by-step Calculation:
Example: 2×2 Matrix
Given Matrix A:
A = [[4, 2],
[2, 3]]
Solution L:
L = [[2.000, 0.000],
[1.000, 1.414]]
Steps:
L[1,1] = √4 = 2
L[2,1] = 2/2 = 1
L[2,2] = √(3-1²) = √2 ≈ 1.414
Algorithm Steps
For diagonal elements:
L[j,j] = √(A[j,j] - Σ(L[j,k]² for k<j))
For lower elements:
L[i,j] = (A[i,j] - Σ(L[i,k]×L[j,k])) / L[j,j]
Upper elements:
L[i,j] = 0 for i < j
Understanding Cholesky Decomposition
What is Cholesky Decomposition?
The Cholesky decomposition factors a symmetric positive definite matrix A into the product A = L·L^T, where L is a lower triangular matrix with positive diagonal entries. This is one of the most important matrix factorizations in numerical linear algebra.
Key Requirements
- •Symmetric: A = A^T (matrix equals its transpose)
- •Positive Definite: All eigenvalues are positive
- •Square: n×n matrix
Applications
Numerical Analysis
Solving linear systems Ax = b efficiently using forward and back substitution
Statistics
Generating samples from multivariate normal distributions and covariance matrix analysis
Optimization
Newton's method for optimization problems and least squares fitting
Mathematical Properties
Uniqueness
For a positive definite matrix, the Cholesky decomposition is unique when L has positive diagonal elements.
Computational Efficiency
Requires ~n³/3 operations, making it twice as fast as LU decomposition for symmetric matrices.
Numerical Stability
Inherently stable without pivoting due to the positive definite property ensuring positive square roots.