Could we have some more diagnostic quantities from `LinearAlgebra.jl` implemented? e.g. `norm()`?

[odunbar]

Hi folks,

Firstly, thanks for the awesome package! I have found delightful use in it to handle collections of storing and using combinations of large structured covariance matrices!

Feature suggestion

It would be nice to have some other diagnostics (particularly norm ) of the type available in LinearAlgebra.jl, also defined for LinearMaps too.

I understand that more more complex operations (e.g. performing SVD ) there are other packages defining this (say, TSVD, or LowRankApprox) but not much of the simple stuff is implemented. Furthermore I think LinearMaps is the right place to do this, because if anyone other package extends these functions to LinearMap objects it is type piracy.

Possible solution for norm

One can define the norm (and probably many diagnostics) of Amap::LinearMap by applying it to a basis Amap*X, and using norms of the mat-vec products.

What we are using right now

An example of norm_linear_map(A,p) that we use is randomly approximates norm(A,p) - In case someone wants a quick solution.

using LinearAlgebra, LinearMaps, Random
function norm_linear_map(A::LM, p::Real = 2; n_eval = nothing, rng = Random.default_rng()) where {LM <: LinearMap}
    m, n = size(A)
    n_basis = isa(n_eval, Nothing) ? n : n_eval
    samples = randn(n, n_basis)  # use unit-normalized gaussian vectors
    for i in 1:size(samples,2)
        samples[:,i] /= norm(samples[:,i])
    end
    out = zeros(m,n_basis) 
    mul!(out,A,samples)
    norm_val = (n/n_basis)^(1/p) * norm(out,p) 
   
    return norm_val
end

but one can also do this deterministically with samples being a full basis, and taking the maximum over the p-norms I think.