Function Repository Resource:

# TakagiDecomposition

Compute the Takagi decomposition of a complex-symmetric matrix

Contributed by: Jan Mangaldan
 ResourceFunction["TakagiDecomposition"][m] gives the Takagi decomposition for a complex-symmetric numerical matrix m as a list of matrices {q,d} where q is a unitary matrix and d is a diagonal matrix.

## Details

The Takagi decomposition is also known as the Autonne-Takagi decomposition.
The matrix m must be square and complex-symmetric (as opposed to Hermitian). That is, m==Transpose[m].
The original matrix m is equal to q.d.Transpose[q].
The diagonal elements of d are real and non-negative.

## Examples

### Basic Examples (3)

Find the Takagi decomposition of a complex-symmetric matrix:

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Confirm the decomposition up to numerical rounding:

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Format q and d:

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### Scope (3)

A complex-symmetric matrix with arbitrary-precision entries:

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Find the Takagi decomposition:

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Confirm the decomposition up to numerical rounding:

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### Properties and Relations (2)

Generate a complex-symmetric matrix:

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Compute its Takagi decomposition:

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q is a unitary matrix:

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d is a diagonal matrix with non-negative entries:

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m is equal to q.d.Transpose[q]:

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For real symmetric positive-definite matrices, TakagiDecomposition gives a result equivalent to SingularValueDecomposition and SchurDecomposition:

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### Possible Issues (1)

TakagiDecomposition only works with approximate numerical matrices:

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## Version History

• 1.0.0 – 13 March 2023