| Function | Description |
|---|
Subspace(basis::Array{<:Number, N+1}) | Subspace from basis, where last index of array indexes the basis elements. |
Subspace(basis::Array{Array{<:Number, N}, 1}) | Subspace from list of basis elements. |
random_subspace(T, dim, size) | Random subspace of the given size and dimension, on base field T. |
random_hermitian_subspace | Random subspace satisfying S == S' |
empty_subspace | Empty subspace |
full_subspace | Full subspace |
| Function | Description |
|---|
size(S) | Size of the elements of S. |
dim(S) | Dimension of subspace S. |
S.basis | Array representing an orthonormal basis of S, with the last index of the array indexing the basis elements. |
each_basis_element(S) | Generator for iterating over orthonormal basis elements. |
random_element(S) | Returns a random element of subspace S. |
perp(S), ~S | Perpendicular (orthogonal) subspace. |
tobasis(S, x) | Transform a vector into basis coordinates. Projects x onto S if x is not an element of S. |
frombasis(S, x) | Returns a vector given the basis coordinates. |
projection(S, x) | Projects vector x onto subspace S. |
hermitian_basis(S) | For a subspace satisfying S == S', returns a basis consisting of Hermitian operators. |
hcat(S1, S2, ...) | Direct sum of subspaces. |
vcat(S1, S2, ...) | Direct sum of subspaces. |
hvcat(rows, S1, S2, ...) | Direct sum of subspaces. |
cat(S1, S2, ...; dims) | Direct sum of subspaces. |
kron(S, T) | Direct product of subspaces. |
adjoint(S), S' | Subspace consisting of adjoints of vectors, $\{ x' : x \in S \}$. |
S ⟂ T | Check whether subspace S is orthogonal to T. |
S == T | Check for equality of subspaces. |
x in S, x ∈ S | Check membership of x in subspace S. |
S ⊆ T | Check whether subspace S is contained in T. |
S ⊇ T | Check whether subspace S contains T. |
S + T, S | T | Linear span of union of subspaces, $\textrm{span}\{ x y : x \in S, y \in T \}$. |
S & T, S ∩ T | Intersection of subspaces. |
S * T | Linear span of products of elements of S and of T. |
S / T | Vector space quotient. Requires T ⊆ S. |
| Function | Description |
|---|
variable_in_space(S) | Creates a variable constrained to space S. |
tobasis(S, x) | Transform variale from vector to basis coordinates. |
frombasis(S, x) | Transform variale from basis coordinates to vector. |
x in S | Creates a constraint requiring variable x to be in subspace S. |
As an example, here is how one could do a positive semidefinite matrix completion.
using Convex, SCS, Subspaces
n = 3
A = [ 1 2 0
0 0 6
0 0 9 ]
basis_element(x, y) = [ i==x && j == y for i in 1:n, j in 1:n ]
S = Subspace([ basis_element(I[1], I[2]) for I in findall(A .== 0) ])
X = variable_in_space(S)
problem = minimize(tr(X), [ X+A ⪰ 0 ])
solve!(problem, () -> SCS.Optimizer(verbose=0, eps=1e-9))
evaluate(X+A)
# output
3×3 Array{Float64,2}:
1.0 2.0 3.0
2.0 4.0 6.0
3.0 6.0 9.0
Alternately, instead of X = variable_in_space(S) we could use X ∈ S.
X = Variable(n, n)
problem = minimize(tr(X), [ X ∈ S, X+A ⪰ 0 ])
solve!(problem, () -> SCS.Optimizer(verbose=0, eps=1e-9))
evaluate(X+A)
# output
3×3 Array{Float64,2}:
1.0 2.0 3.0
2.0 4.0 6.0
3.0 6.0 9.0