Function Repository Resource:

# FullRankDecomposition

Compute a full rank decomposition of a matrix

Contributed by: Jan Mangaldan
 ResourceFunction["FullRankDecomposition"][m] computes a full rank decomposition of the matrix m.

## Details and Options

The result is given in the form {f,g} where f and g are matrices such that f.g==m.
In a full rank decomposition, the matrix m of dimension p×q is factored into a matrix f with dimensions p×r and a matrix g with dimensions r×q, where r is equal to MatrixRank[m].
The full rank decomposition is not unique; if f1.g1==m and f2.g2==m are two different decompositions of m, then there exists a matrix w with dimensions r×r such that f1==f2.w and w.g1==g2.
ResourceFunction["FullRankDecomposition"] takes the same options as RowReduce.

## Examples

### Basic Examples (3)

Find a full rank decomposition of a rank-deficient matrix:

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Confirm the decomposition:

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Format the factors:

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### Scope (2)

Find a full rank decomposition of a rectangular matrix with exact entries:

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Confirm the decomposition:

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Find a full rank decomposition of a complex matrix with inexact entries:

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Confirm the decomposition up to numerical rounding:

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## Version History

• 1.0.0 – 25 January 2023