Linear Subspaces

Coordinates

A type used for encoding the used coordinates and for performing coordinate changes.

Currently supported coordinates are:

• Intrinsic
• Extrinsic

Intrinsic <: Coordinates

Indicates the use of the intrinsic description of an (affine) linear subspace. See also IntrinsicDescription.

Extrinsic <: Coordinates

Indicates the use of the extrinsic description of an (affine) linear subspace. See also ExtrinsicDescription.

LinearSubspace(A, b)

An $m$-dimensional (affine) linear subspace $L$ in $n$-dimensional space given by the extrinsic description $L = \{ x | A x = b \}$.

julia> A = LinearSubspace([1 0 3; 2 1 3], [5, -2])
1-dim. (affine) linear subspace {x|Ax=b} with eltype Float64:
A:
2×3 Array{Float64,2}:
 1.0  0.0  3.0
 2.0  1.0  3.0
b:
2-element Array{Float64,1}:
  5.0
 -2.0

julia> dim(A)
1

julia> codim(A)
2

julia> ambient_dim(A)
3

A LinearSubspace holds always its extrinsic description, see also ExtrinsicDescription, as well as its intrinsic description, see IntrinsicDescription.

julia> intrinsic(A)
IntrinsicDescription{Float64}:
A:
3×1 Array{Float64,2}:
 -0.6882472016116853
  0.6882472016116853
  0.22941573387056186
b:
3-element Array{Float64,1}:
 -3.0526315789473677
 -3.947368421052632
  2.684210526315789

A LinearSubspace can be evaluated with either using Intrinsic or Extrinsic coordinates.

julia> u = [0.5]
1-element Array{Float64,1}:
 0.5

julia> x = A(u, Intrinsic)
3-element Array{Float64,1}:
 -3.3967551797532103
 -3.6032448202467893
  2.79891839325107

julia> A(x, Extrinsic)
  2-element Array{Float64,1}:
   0.0
   0.0

To change the used coordinates you can use coord_change.

julia> coord_change(A, Extrinsic, Intrinsic, x)
1-element Array{Float64,1}:
 0.49999999999999994

ExtrinsicDescription(A, b)

Extrinsic description of an $m$-dimensional (affine) linear subspace $L$ in $n$-dimensional space. That is $L = \{ x | A x = b \}$. Note that internally A and b will be stored such that the rows of A are orthonormal.

IntrinsicDescription(A, b)

Intrinsic description of an $m$-dimensional (affine) linear subspace $L$ in $n$-dimensional space. That is $L = \{ u | A u + b \}$. Here, $A$ and $b$ are in orthogonal coordinates. That is, the columns of $A$ are orthonormal and $A' b = 0$.

ambient_dim(A::LinearSubspace)

Dimension of ambient space of the (affine) linear subspace A.

codim(A::ExtrinsicDescription)

Codimension of the (affine) linear subspace A.

codim(A::IntrinsicDescription)

Codimension of the (affine) linear subspace A.

codim(A::LinearSubspace)

Codimension of the (affine) linear subspace A.

codim(W::WitnessSet)

The dimension of the algebraic set encoded by the witness set.

coord_change(A::LinearSubspace, C₁::Coordinates, C₂::Coordinates, p)

Given an (affine) linear subspace A and a point p in coordinates C₁ compute the point x describing p in coordinates C₂.

Example

julia> A = LinearSubspace([1 0 3; 2 1 3], [5, -2]);

julia> u = [1.25];

julia> x = coord_change(A, Intrinsic, Extrinsic, u)
3-element Array{Float64,1}:
 -3.9129405809619744
 -3.087059419038025
  2.9709801936539915

julia> A(x, Extrinsic)
2-element Array{Float64,1}:
 0.0
 0.0

julia> x - A(u, Intrinsic)
3-element Array{Float64,1}:
 0.0
 0.0
 0.0

dim(A::ExtrinsicDescription)

Dimension of the (affine) linear subspace A.

dim(A::IntrinsicDescription)

Dimension of the (affine) linear subspace A.

dim(A::LinearSubspace)

Dimension of the (affine) linear subspace A.

dim(W::WitnessSet)

The dimension of the algebraic set encoded by the witness set.

intrinsic(A::LinearSubspace)

Obtain the intrinsic description of A, see also IntrinsicDescription.

is_linear(L::LinearSubspace)

Returns true if the space is proper linear subspace, i.e., described by `L = { x | Ax = 0 }.

extrinsic(A::LinearSubspace)

Obtain the extrinsic description of A, see also ExtrinsicDescription.

geodesic(V::LinearSubspace, W::LinearSubspace)

Compute the geodesic distance between `V = {x | Ax = a}` and `W= {x | Bx = b}` as `sqrt(d^2 + ||a-b||^2)`, where `d` is the distance from the columnspan of `A` to the columnspan of `B` in the Grassmannian. This follows the derivation in [^LKK19].

Returns the geodesic $γ(t)$ by connecting V = {x | Ax = a} and W= {x | Bx = b}. A and B are interpolated in the Grassmannian using Stiefel coordinates. See also Corollary 4.3 in ^[LKK19]. a and b are interpolated linearly. Thus, $γ(t)$ is a geodesic in the Euclidean group.

geodesic_distance(V::LinearSubspace, W::LinearSubspace)

Compute the geodesic distance between V = {x | Ax = a} and W= {x | Bx = b} as sqrt(d^2 + ||a-b||^2), where d is the distance from the columnspan of A to the columnspan of B in the Grassmannian. This follows the derivation in ^[LKK19].

rand_subspace(n::Integer; dim | codim, affine = true, real = false)

Generate a random LinearSubspace with given dimension dim or codimension codim (one of them has to be provided) in ambient space of dimension n. If real is true, then the extrinsic description is real. If affine then an affine linear subspace is generated. The subspace is generated by drawing the matrix A the extrinsic description independently from a normal distribuation using randn. The offset is drawn from the normal distribution scaled by sqrt(n), where n is the number of columns of A.

rand_subspace(x::AbstractVector; dim | codim, affine = true)

Generate a random LinearSubspace with given dimension dim or codimension codim (one of them has to be provided) in ambient space of dimension length(x) going through the given point x.

Example

Construction of a general random subspace:

julia> rand_subspace(3; dim = 1)
1-dim. affine linear subspace {x | Ax=b} with eltype ComplexF64:
A:
ComplexF64[-0.18709056988162928 - 0.021159068437462684im 0.3448082517062497 - 0.5619954950326371im 0.12814427601487394 + 0.716517124796854im; 0.6274330537397367 + 0.4327848781661207im 0.3433265716931795 - 0.4388117939096428im -0.09290449999579178 - 0.3161721696818136im]
b:
ComplexF64[0.1568834271069519 - 0.7893604951164849im, -1.068448434670914 + 3.422846499360692im]

julia> rand_subspace(4; codim = 1)
3-dim. affine linear subspace {x | Ax=b} with eltype ComplexF64:
A:
ComplexF64[-0.16031107752943963 - 0.6372670776364352im 0.4895718108240883 + 0.4093061519132399im -0.32944285959791164 + 0.12384710749057845im -0.17790282756300982 - 0.07388387107969129im]
b:
ComplexF64[-0.09372015908519082 + 0.3051873249600184im]

translate(L::LinearSubspace, δb, ::Coordinates = Extrinsic)

Translate the (affine) linear subspace L by δb.

Base.intersect(L₁::LinearSubspace, L₂::LinearSubspace)

Intersect the two given linear subspaces. Throws an ErrorException if the intersection is the sum of the codimensions is larger than the ambient dimension.