Home > Cholesky decomposition

1 Definition

Let A be a positive definite square matrix over a field, then A can be decomposed as

with L a lower triangular matrix with positive diagonal entries, and L^* the conjugate transpose of L.

2 Notes

If the matrix has only real valued entries the conjugate transpose coincides with the transpose and the decomposition simplifies to

2.1 Cholesky algorithm

The Cholesky algorithm, used to calculate the decomposition matrix L, is a modified version of the Gauss algorithm.

The recursive algorithm start with

We define

thus

The recursion terminates after n-steps when A⁽ⁿ⁾ = 1. The lower triangle matrix L we are looking for is calculated as

2.2 Cholesky Banachiewicz algorithm

The Cholesky Banachiewicz algorithm gives a formula to calculate the entries of the lower triangle matrix L directly. It starts form the upper left corner of the matrix L and proceeds to calculate the matrix row by row.

for i = 1,...,m

for j = 1,...,i

2.3 Cholesky Crout algorithm

The Cholesky Crout algorithm provides a slightly different way to calculate the entries of the lower triangle matrix L. It starts from the upper left corner of the matrix L and proceeds to calculate the matrix column by column.

for i = 1,...,m

for j = i,...,m

3 See also

• symbolic Cholesky decomposition
• minimum degree algorithm