Reference

PLUQ decomposition

RankRevealing.PLUQ — Type

PLUQ <: Factorizarion

Matrix factorization type for the full LU factorization of any rectangular matrix A over an arbitrary numeric field. This is the return type of pluq.

The factorization satisfies

P * [L ; M] * [U V] * Q == A

Component	Description
`F.L`	Full-rank unit lower triangular part
`F.U`	Full-rank upper triangular part
`F.M`	Remainder of the row dimension
`F.V`	Remainder of the column dimension
`F.P`	row-wise permutation `Matrix`
`F.Q`	column-wise permutation `Matrix`
`F.p`	row-wise permutation `Vector`
`F.q`	column-wise permutation `Matrix`

Iteration and destructuring produce the components in the order p, L, U, V, M, q`.

Both L and U are square full-rank matrices with the same rank as A.

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RankRevealing.pluq — Function

pluq(A) -> PLUQ

Perform a rank-sensitive LU factorization of A. The factorization of an n by m matrix is computed in O(n m r^(ω-2)) steps, where r = rank(A) and ω is the matrix multiplication complexity exponent.

This is an exact function who should work with any type implementing the basic arithmetic operations +, -, *, /, one, zero.

The resulting factorization F satisfies

F.P * [F.L ; F.M] * [F.U F.V] * F.Q == A

See the link for the original description of the algorithm.

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Generalized Rank Revealing decomposition

RankRevealing.GeneralizedRankRevealing — Type

GeneralizedRankRevealing <: Factorization

Matrix factorization type for the Generalized Rank Reveling decomposition of two matrices A and B with the same number of columns. This is obtained via the function grr(A,B).

By calling d_+ the dimension of the sum of A and B row spaces, this factorization has the decomposes the matrices as

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

Component	Description
`F.X`	`m_A x r_A` full column rank matrix
`F.Y`	`m_B x r_B` full column rank matrix
`F.H`	`d_+ x n` full row rank matrix

Other useful fields are:

F.r1: Rank of A minus minus the intersection
F.r2: Rank of B minus minus the intersection
F.r3: Rank of the intersection

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RankRevealing.grr — Function

grr(A, B) -> GeneralizedRankRevealing

Compute the Generalized LU decomposition of matrices A and B. There are matrices X, Y and H such that

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

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RankRevealing.simple_rr — Function

simple_rr(A)

Decompose A == X*H where X is full column rank and H is full row rank.

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RankRevealing.left_rr — Function

left_rr(A)

Decompose A == X*[I ; 0]*H where X is invertible and H is full row rank.

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RankRevealing.right_rr — Function

right_rr(A)

Decompose A == X*[0 I]*H where X is full column rank and H is invertible.

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Working with subspaces

RankRevealing.basis — Function

basis(xs)

Turn a set of vectors into a set of linearly independet vectors spawning the same subspace.

Warning: Even if the input is already linearly independent, the returned basis may be composed of other vectors spawning the same subspace.

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RankRevealing.intersection — Function

intersection(xs, ys) -> vs

Given two sets of vectors, calculate a basis for the intersection of the subspaces generated by them.

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RankRevealing.sumspace — Function

sumspace(xs, ys) -> vs

Given two sets of vectors, calculate a basis for the sum space of the subspaces generated by them.

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