Function Repository Resource:

LDLDecomposition

Find the LDL decomposition of a Hermitian matrix

Contributed by: Jan Mangaldan and Daniel Lichtblau

ResourceFunction["LDLDecomposition"][mat]

finds the LDL decomposition of Hermitian matrix mat.

Details

For a Hermitian matrix mat, the LDL decomposition is a pair {lower,diag} such that lower is lower diagonal matrix with ones on the main diagonal, diag is a diagonal matrix and mat=lower.diag.ConjugateTranspose[lower].

When mat is positive definite the LDL decomposition is a simple modification of CholeskyDecomposition. In this case it is equivalent to the resource function RationalCholeskyDecomposition.

Examples

Basic Examples (5)

Create a Hankel matrix:

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Check that it is Hermitian:

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Compute the LDL decomposition:

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Check that these matrices are lower triangular with ones on the main diagonal, and diagonal respectively:

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Check that we recover the original hmat from the LDL product:

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Scope (3)

LDLDecomposition works with complex Hermitian matrices:

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Compute the LDL decomposition:

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Check that these matrices have the required properties:

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Check that we recover the original matrix:

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LDLDecomposition works with exact numeric matrices:

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Compute the LDL decomposition:

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Check that these matrices have the required properties:

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Check that we recover the original matrix:

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LDLDecomposition works with symbolic matrices provided it can determine that they are Hermitian:

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Check that mat is Hermitian:

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Compute the LDL decomposition:

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Check that we recover the original matrix:

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Properties and Relations (2)

Create a Hilbert matrix of dimension 5:

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Check that it is positive definite:

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Compute the LDL decomposition:

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Check that this agrees with the rational Cholesky decomposition up to numeric machine precision error:

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{l2, d2} = ResourceFunction["RationalCholeskyDecomposition"][hilb];
d2 = DiagonalMatrix[d2];
Chop[{l1 - l2, d1 - d2}]

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For positive definite matrices one can obtain the LDL decomposition from the LU decomposition (when no pivoting was performed) and also from the Cholesky decomposition. Form a positive definite matrix:

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