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# CholeskyDecomposition

 gives the Cholesky decomposition of a matrix m.
• The matrix m can be numerical or symbolic, but must be Hermitian and positive definite.  »
• yields an upper-triangular matrix u so that ConjugateTranspose[u].um.  »
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:
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:
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:
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:
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