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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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