Function Repository Resource:

# SkewLTLDecomposition

Tridiagonalize an antisymmetric (skew-symmetric) matrix using the Parlett-Reid algorithm

Contributed by: Wolfram Staff (original content by M. Wimmer)
 ResourceFunction["SkewLTLDecomposition"][m] gives the Parlett–Reid decomposition of the skew-symmetric matrix m.

## Details and Options

The result is given in the form {l,t,p}, where l is a lower triangular matrix with a unit diagonal, t is a tridiagonal matrix and p a permutation matrix such that p.m.pTl.t.lT.
Skew-symmetric matrices are also called antisymmetric.
ResourceFunction["SkewLTLDecomposition"] can be considered a generalization of the LUDecomposition.

## Examples

### Basic Examples (1)

The Parlett–Reid decomposition of a skew-symmetric matrix:

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### Scope (3)

The Parlett–Reid decomposition of a real antisymmetric matrix:

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Decomposing of a complex antisymmetric matrix:

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SkewLTLDecomposition applied to a symbolic antisymmetric matrix:

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### Properties and Relations (3)

Compute the Pfaffian of an antisymmetric matrix by reducing it to the tridiagonal form:

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In the result of {l,t,p}=SkewLTLDecomposition[m], the matrix l is lower-triangular with a unit diagonal and t is tridiagonal:

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The product is given by p.m.pT:

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The resource function SkewTridiagonalDecomposition also produces a tridiagonal matrix t with the same Pfaffian, possibly up to the sign:

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Use the resource function Pfaffian to make the comparison:

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## Version History

• 1.0.0 – 04 November 2020