CholeskyDecomposition

CholeskyDecomposition[m]

gives the Cholesky decomposition of a matrix m.

Details

Examples

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Basic Examples  (1)

The matrix is positive definite:

Scope  (2)

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:

Generalizations & Extensions  (1)

Use symbolic matrices:

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

Applications  (1)

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:

Properties & Relations  (2)

m is a symmetric positive definite matrix:

Compute the Cholesky decomposition:

Verify ConjugateTranspose[u].u == m:

m is a random matrix with real entries:

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

Find the QRDecomposition of m:

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

Possible Issues  (2)

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:

Introduced in 2003
 (5.0)