Built-in Mathematica Symbol

CholeskyDecomposition

CholeskyDecomposition[m]
gives the Cholesky decomposition of a matrix m.

The matrix m can be numerical or symbolic, but must be Hermitian and positive definite. »

CholeskyDecomposition[m] yields an upper-triangular matrix u so that ConjugateTranspose[u].um. »

The matrix is positive definite:

Hilbert matrices are symmetric and positive definite:

Compute the Cholesky decomposition with exact arithmetic:

Compute the Cholesky decomposition with machine arithmetic:

Compute the Cholesky decomposition with 24-digit precision arithmetic:

Compute the Cholesky decomposition of a random complex Hermitian matrix:

Use symbolic matrices:

Conditions are needed to make sure the matrix is positive definite:

The Cholesky decomposition is a fast way of determining positive definiteness:

The identity matrix is positive definite:

Estimate the probability that

is positive definite for r, a random 3×3 matrix:

m is a symmetric positive definite matrix:

Compute the Cholesky decomposition:

m:

m is a random matrix with real entries:

Find the Cholesky decomposition of Transpose[m].m:

r is the same as u except for the choice of sign for each row:

Matrices need to be positive definite enough to overcome numerical roundoff:

The smallest eigenvalue is effectively zero to machine precision:

The decomposition can be computed when the precision is high enough to resolve it:

s is a sparse tridiagonal matrix:

The Cholesky decomposition is computed as a dense matrix even if the result is sparse:

Using LinearSolve will give a LinearSolveFunction that has a sparse Cholesky factorization: