Cholesky decomposition

1 Cholesky Decomposition

A symmetricPlanetmathPlanetmath and positive definite matrix can be efficiently decomposed into a lower and upper triangular matrixMathworldPlanetmath. For a matrix of any type, this is achieved by the LU decompositionMathworldPlanetmath which factorizes A=LU. If A satisfies the above criteria, one can decompose more efficiently into A=LLT where L is a lower triangular matrix with positive diagonal elements. L is called the Cholesky triangle.

To solve Ax=b, one solves first Ly=b for y, and then LTx=y for x.

A variant of the Cholesky decomposition is the form A=RTR , where R is upper triangular.

Cholesky decomposition is often used to solve the normal equationsMathworldPlanetmath in linear least squares problems; they give ATAx=ATb , in which ATA is symmetric and positive definitePlanetmathPlanetmath.

To derive A=LLT, we simply equate coefficients on both sides of the equation:


Solving for the unknowns (the nonzero ljis), for i=1,,n and j=i-1,,n, we get:

lii = (aii-k=1i-1lik2)
lji = (aji-k=1i-1ljklik)/lii

Because A is symmetric and positive definite, the expression under the square root is always positive, and all lij are real.


  • 1 Originally from The Data Analysis Briefbook (
Title Cholesky decomposition
Canonical name CholeskyDecomposition
Date of creation 2013-03-22 12:07:38
Last modified on 2013-03-22 12:07:38
Owner gufotta (12050)
Last modified by gufotta (12050)
Numerical id 16
Author gufotta (12050)
Entry type Definition
Classification msc 62J05
Classification msc 65-00
Classification msc 15-00
Synonym Cholesky factorization
Synonym matrix square root
Related topic SquareRootOfPositiveDefiniteMatrix
Defines Cholesky triangle