SymmetricMatrixCholeskyDecomposition Method

public CholeskyDecomposition CholeskyDecomposition()

Public Function CholeskyDecomposition As CholeskyDecomposition

public:
CholeskyDecomposition^ CholeskyDecomposition()

member CholeskyDecomposition : unit -> CholeskyDecomposition 

Return Value

A Cholesky decomposition is a special decomposition that is possible only for positive definite matrices. (A positive definite matrix M has x^TMx > 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(N³) 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.

Reference