Function Repository Resource:

# AasenDecomposition

Compute the Aasen decomposition of a Hermitian matrix

Contributed by: Jan Mangaldan
 ResourceFunction["AasenDecomposition"][m] yields the Aasen decomposition of a matrix m.

## Details and Options

The matrix m can be numerical or symbolic, but must be Hermitian.
The result of ResourceFunction["AasenDecomposition"] is a list {l,t,p}, where l is a unit lower triangular matrix, t is a SparseArray representing a Hermitian tridiagonal matrix, and p is a permutation matrix.
The matrices returned by ResourceFunction["AasenDecomposition"] satisfy the relation p.m.Transpose[p]=l.t.ConjugateTranspose[l].
With the setting TargetStructure"Structured", ResourceFunction["AasenDecomposition"][m] returns a list {l,t,p} where l is a LowerTriangularMatrix, t is a SparseArray representing a Hermitian tridiagonal matrix, and p is a PermutationMatrix.

## Examples

### Basic Examples (3)

Find the Aasen decomposition of a symmetric matrix:

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View the results in matrix form:

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Verify the decomposition:

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### Scope (3)

A symmetric indefinite matrix:

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Compute the Aasen decomposition with exact arithmetic:

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Compute the Aasen decomposition with machine arithmetic:

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Compute the Aasen decomposition with 24-digit precision arithmetic:

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Generate a pseudorandom Hermitian matrix drawn from the Gaussian unitary ensemble:

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Compute its Aasen decomposition:

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Verify the decomposition:

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Compute the Aasen decomposition of a symbolic matrix:

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### Options (1)

#### TargetStructure (1)

With TargetStructure"Structured", a list containing a LowerTriangularMatrix, a SparseArray representing a Hermitian tridiagonal matrix, and a PermutationMatrix is returned:

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## Version History

• 3.0.0 – 27 March 2024
• 2.0.0 – 29 August 2022
• 1.0.0 – 11 February 2021