Add `isunitary` & `isorthogonal` functions to `LinearAlgebra` stdlib?

[singularitti]

Would that be possible to do it? We already have something like ishermitian.

[singularitti]

I could help?

[jishnub]

@PredictiveManish There's no formal assignment. You may submit a PR with the implementations added, along with appropriate tests.

I wonder if there is an easy way to check this without verifying U*U' == I to within some tolerance? Checking for symmetry is O(N^2) as only the elements need to be compared.

[barucden]

Is it faster to compute the multiplication U * U' than looping through the columns and just checking dot(u, v) = 0 (or 1)? The latter could return early in the case of non-orthogonal matrices (plus it would not allocate).

Anyway, for certain matrices (e.g., Diagonal), the check can be simpler, right?

[dkarrasch]

I'm not sure how these functions would be used. Symmetry (or Hermitian-ness) is checked exactly, based on the components, and we don't provide an "almost-symmetric check", for good reasons, because bad things can happen when you perturb symmetric matrices out of symmetric matrices. But what do you do with a result like "is orthogonal up to an error"? So it's "not orthogonal", or "almost orthogonal"? I believe somebody has requested adding an orthogonality check function for vectors, but again what do you do with "numerically almost orthogonal vectors"?

It's something different when you have an object that comes from an algorithm that is theoretically guaranteed to return orthogonal objects (like columns of AbstractQ operators, or AbstractQs themselves), but checking random matrices or vectors for "sort of orthogonal"? I'm not sure.

I think one would need more background and a very good reason to add such fragile functions, otherwise it could very well be implemented in a few lines in a package.

[singularitti]

I've thinking about that too, @dkarrasch. These functions are implemented in https://github.com/jlapeyre/IsApprox.jl.

[aravindh-krishnamoorthy]

I think one would need more background and a very good reason to add such fragile functions, otherwise it could very well be implemented in a few lines in a package.

I agree with this.

Furthermore, istriangular(.), ishermitian(.), and such are necessary in STDLIB because they are used to identify, e.g., the LAPACK routines to be called. For now, I don't see routines that explicitly require unitary or orthogonal inputs. Hence, it may be best to provide this functionality in a separate package.

[singularitti]

Maybe the others are not so easy to implement, but we can add is_skew_symmetric just like issymmetric without any issue? Since it is pretty easy to verify: $A^\intercal = -A$.

[stevengj]

@singularitti, the SkewLinearAlgebra.jl package provides an isskewhermitian(A) function (and lots of functions to take advantage of skew-Hermitian matrices).

In LinearAlgebra.jl, checking for skew-symmetric matrices seems pretty useless since we have no functions that take advantage of it?

One place that might be useful is in exp … a lot of people (e.g. coming from quantum mechanics) do exp(im*H) where H is Hermitian, not realizing that it is probably much more efficient in that case to use cis(H). It might be worth checking for this case in the exp function.