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Cholesky Decomposition

Usage

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

Basic examples

Compute the Cholesky decomposition of a 2×2 real matrix:
In[1]:=CholeskyDecomposition[(2 | 1
1 | 2)]
Out[1]={{sqrt(2), 1/sqrt(2)}, {0, sqrt(3/2)}}
Verify the decomposition:
In[2]:=ConjugateTranspose[%].%//MatrixForm
Out[2]=(2 | 1
1 | 2)
The original matrix is positive definite:
In[3]:=PositiveDefiniteMatrixQ[(2 | 1
1 | 2)]
Out[3]=True
Compute the Cholesky decomposition of a 3×3 complex Hermitian matrix:
In[1]:=CholeskyDecomposition[(2.41 + 0. i | -2.47 - 1.92 i | 1.11 + 2.2 i
-2.47 + 1.92 i | 6.25 + 0. i | -4.12 - 1.35 i
1.11 - 2.2 i | -4.12 + 1.35 i | 3.43 + 0. i)]//MatrixForm
Out[1]=(1.55366 + 0. i | -1.58688 - 1.23724 i | 0.714598 + 1.41639 i
0. + 0. i | 1.48414 + 0. i | -0.829623 + 0.010836 i
0. + 0. i | 0. + 0. i | 0.478266 + 0. i)
The result is upper triangular:
In[2]:=UpperTriangularMatrixQ[%]
Out[2]=True

Option

TargetStructure

Relationships with other entities

Andre-Louis Cholesky

LUDecomposition | LinearSolve | LinearSolveFunction | FindMinimum | PseudoInverse | QRDecomposition | HermitianMatrixQ | PositiveDefiniteMatrixQ

Relationships with other entities

History

introduced in Version 5 (June 2003)
last modified in Version 14 (December 2023)

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