Meta.Numerics Library

## SymmetricMatrixCholeskyDecomposition Method |

Computes the Cholesky decomposition of the matrix.

Syntax

The Cholesky decomposition of the matrix, or null if the matrix is not positive definite.

Remarks

A Cholesky decomposition is a special decomposition that is possible only for positive definite matrices.
(A positive definite matrix M has x^{T}Mx > 0 for any vector x. Equivilently, M is positive definite if
all its eigenvalues are positive.)

The Cholesky decomposition represents M = C C^{T}, where C is lower-left triangular (and thus C^{T}
is upper-right triangular. It is basically an LU decomposition where the L and U factors are related by transposition.
Since the M is produced by multiplying C "by itself", the matrix C is sometimes call the "square root" of M.

Cholesky decomposition is an O(N^{3}) operation. It is about a factor of two faster than LU decomposition,
so it is a faster way to obtain inverses, determinates, etc. if you know that M is positive definite.

The fastest way to test whether your matrix is positive definite is attempt a Cholesky decomposition. If this method returns null, M is not positive definite.

See Also