Rank Revealing Decompositions

This package defines methods for exact linear algebra over any numerical field. Our focus is on matrix decompositions that are rank-sensitive. That is, faster for low-rank matrices.

Documentation	Build Status

This package is not registered on Julia's general registry yet. But you can install it by passing the repositories' url to the package manager.

To instal, you have to enter ] on the Julia REPL and write

pkg> add https://github.com/iagoleal/RankRevealing.jl

This a version of the LU decomposition that works for rectangular and possible singular matrices. In block notation, it turns a matrix A into

A = P * |L| * | U V | * Q
        |M|

where P, Q are permutations, L is non-singular unit lower triangular, and U is non-singular upper triangular. Also, the dimensions of L and U are equal to the rank of A.

If A is m x n and has rank r, the PLUQ takes O(m n r^(ω-2)) steps, where ω is the exponent for the square matrix multiplication algorithm used by Julia.

See [1] for the paper this implementation is based.

Given two matrices A and B with the same number of columns, this decomposition divides the space based on how their row spaces interact. In block notation, it decomposes the matrices as

                | I 0 0 |
| A | = | X 0 | | 0 0 I |
| B |   | 0 Y | | 0 I 0 | H
                | 0 0 I |

Also, the rows of H form bases for

• R(A), the row space of A,
• R(B), the row space of B,
• R(B) ∩ R(B), the row spaces intersection,
• R(B) + R(B), the sum of their row spaces,

with the additional property that whenever one of those spaces is contained in another, the calculated bases share the necessary vectors.

If A is m_A x n, B is m_B x n and d is the dimension of their images' sum space ({ a + b | ∃x, y, a = xA, b = yB }), then the GRR decomposition takes O((m_A + m_B) n d^(ω-2)) steps, where ω is the exponent for the square matrix multiplication algorithm used by Julia.

• [1] Jean-Guillaume Dumas, Clément Pernet, Ziad Sultan. "Simultaneous computation of the row and column rank profiles". In: Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation. 2013, pp. 181–188.

• [2] Iago Leal de Freitas, João Paixão, Lucas Rufino, and Paweł Sobocínski. "Rank sensitive complexity to find the intersection between two subspaces". 2022 (upcoming)