`right_null` for a simple matrix

[Yue-Zhengyuan]

MWE (using the latest main branch):

using MatrixAlgebraKit, LinearAlgebra
# a very simple rank-1 2x2 matrix
A = [1. 0.; 1. 0.]
# it has a 1D (right) nullspace, spanned by [0, 1]
N = nullspace(A)
#= output
2×1 Matrix{Float64}:
 0.0
 1.0
=#

# MAK fails to find it, with either alg = :svd or :qr
right_null(A)
#= output
0×2 Matrix{Float64}
=#

For this simple case, the behavior does not match the docstring:

Compute an orthonormal basis N = adjoint(Nᴴ) for the kernel of the matrix A, 
i.e. the nullspace of A, such that A * Nᴴ' ≈ 0 and Nᴴ * Nᴴ' ≈ I.

[lkdvos]

I think what you are finding here is some weird quirk with how the interface works.
By default, the QR or LQ nullspace is computed, which isn't rank-revealing so only nullspaces of rectangular matrices can be found with that.
Then, for SVD this does work, however the default setting is trunc = nothing, which effectively means that again only rectangular matrices can be detected.
As soon as you actually specify a tolerance though, either through trunc = (; atol = 1e-12) or trunc = trunctol(; ...), the nullspace is found:

julia> right_null(A)
0×2 Matrix{Float64}

julia> right_null(A; alg = :svd)
0×2 Matrix{Float64}

julia> right_null(A; trunc = (; atol = 1e-12))
1×2 Matrix{Float64}:
 0.0  1.0

I do agree that this might not be the most intuitive default settings, but I'm not entirely sure how else to set them.
If you have suggestions, definitely welcome!

[Yue-Zhengyuan]

Then at least some warning should be added to the docstring about the default QR/LQ algorithm.

I took a look at LinearAlgebra.nullspace. It is using the SVD algorithm, with the following truncation setting:

function nullspace(A::AbstractVecOrMat; atol::Real=0, rtol::Real = (min(size(A, 1), size(A, 2))*eps(real(float(oneunit(eltype(A))))))*iszero(atol))

In its docstring (M refers to A):

"""
Computes a basis for the nullspace of `M` by including the singular
vectors of `M` whose singular values have magnitudes smaller than `max(atol, rtol*σ₁)`,
where `σ₁` is `M`'s largest singular value.

By default, the relative tolerance `rtol` is `n*ϵ`, where `n`
is the size of the smallest dimension of `M`, and `ϵ` is the [`eps`](@ref) of
the element type of `M`.
"""

So you may consider adopting the same atol, rtol as well. But I also saw you write, "should keep exact zeros due to rectangularity", which I don't yet understand the reason.