Function Repository Resource:

# SkewTridiagonalDecomposition

Compute the skew-tridiagonal decomposition of an antisymmetric matrix

Contributed by: Wolfram Staff (original content by M. Wimmer)
 ResourceFunction["SkewTridiagonalDecomposition"][m] gives the skew-tridiagonal decomposition of antisymmetric matrix m.

## Details and Options

The result is given in the form {q,t}, where q is a unitary matrix and t is a tridiagonal matrix such that mq.t.qT.
Skew-symmetric matrices are also called antisymmetric.
ResourceFunction["SkewTridiagonalDecomposition"] is computed using Householder transformations.

## Examples

### Basic Examples (2)

Construct a a skew-symmetric matrix:

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The skew-tridiagonal decomposition:

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### Scope (2)

The skew-tridiagonal decomposition of a real antisymmetric matrix:

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The skew-tridiagonal decomposition of a complex antisymmetric matrix:

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### Properties and Relations (4)

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

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Compare with the result of the resource function Pfaffian:

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In the result of {q,t}=SkewTridiagonalDecomposition[m], the matrix q is unitary and t is tridiagonal:

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The original matrix is given by q.t.Transpose[q]:

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For real matrices, SkewTridiagonalDecomposition gives result similar to HessenbergDecomposition:

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

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## Version History

• 1.0.0 – 04 November 2020