Cholesky Decomposition
Free online Cholesky decomposition calculator for symmetric positive definite 2x2 matrices. Computes the lower triangular factor L and verifies A = LL^T.
For symmetric positive definite matrix A = [[a, b], [b, d]]
About This Calculator
The Cholesky decomposition factors a symmetric positive definite matrix A into the product of a lower triangular matrix L and its transpose: A = LLT. For a 2×2 matrix A = [[a, b], [b, d]], the Cholesky factor L = [[l11, 0], [l21, l22]] is computed as:
- l11 = √a
- l21 = b / l11
- l22 = √(d − l212)
This decomposition is twice as efficient as LU decomposition for symmetric positive definite matrices and is the method of choice in many numerical applications including solving linear systems, Monte Carlo simulations, and Kalman filtering.
Frequently Asked Questions
How does the Cholesky decomposition work?
Cholesky decomposition factors a symmetric positive definite matrix A into the product of a lower triangular matrix L and its transpose: A = LLT. For a 2×2 matrix, it computes L by taking the square root of the first pivot element, then solving for the remaining entries. It is named after André-Louis Cholesky, a French military officer and mathematician.
What is a symmetric positive definite matrix?
A symmetric positive definite matrix is a square matrix that equals its own transpose and has all positive eigenvalues. In the 2×2 case with entries [[a, b], [b, d]], the conditions are a > 0 and ad − b² > 0. These matrices arise naturally in least-squares problems, optimization, and physics simulations.
What are the applications of Cholesky decomposition?
Cholesky decomposition is widely used in numerical linear algebra for solving systems of linear equations efficiently, Monte Carlo simulations in finance, Kalman filters in control theory, nonlinear optimization, and Gaussian process regression in machine learning. It is roughly twice as fast as LU decomposition for symmetric positive definite matrices.
Is this tool free?
Yes, all calculators on Calculy are completely free to use. There are no hidden fees, subscriptions, or usage limits. You can compute as many matrix decompositions as you need.
When does Cholesky decomposition fail?
Cholesky decomposition fails when the matrix is not symmetric positive definite. Specifically, it fails if the first element a ≤ 0 or if the determinant ad − b² ≤ 0. If the matrix is only slightly non-positive-definite due to rounding errors, modified Cholesky algorithms can be used instead.
How is Cholesky different from LU decomposition?
LU decomposition works for any invertible square matrix, while Cholesky requires the matrix to be symmetric positive definite. Cholesky is approximately twice as efficient as LU because it exploits the matrix symmetry. Cholesky produces L where A = LLT, whereas LU produces separate lower (L) and upper (U) factors where PA = LU.