Change the behavior of dot to not conjugate, and introduce an inner (product) function

[jebej]

As briefly discussed in #22220 (with relevance to #20978), I suggest removing the conjugating behavior from dot, and introducing an inner function which does take the adjoint. This is consistent with numpy. The dot notation seems to rarely be used for complex numbers, for which the general inner product terminology ⟨a,b⟩ is more common. Of course, for real numbers, we have inner == dot.

I'm not sure where the discussion concerning the recursiveness of the dot product should go. It's been mentioned in #20978, but that doesn't seem to be the main point of that issue.

[stevengj]

This seems wrong to me (and will be breaking). dot should denote an inner product. (The recursiveness is essential for it to be an inner product.)

If you want an unconjugated sum, we could define dotu analogous to BLAS. (In fact, we already have a non-exported dotu function.)

[stevengj]

See also #8300.

[jebej]

I suppose the name change is not too important, especially with how breaking it would be. I still think inner is more appropriate a name however. I guess I can do const inner = dot if I really want to.

(The recursiveness is essential for it to be an inner product.)

I'm not an expert, but I'd be curious to learn more about this. The only required properties of an inner product space are conjugate symmetry, linearity and positive-definiteness. Would a non-recursive inner product not satisfy those properties?

[stevengj]

An inner product for a vector space needs to give an element of the scalar field. Since scalar multiplication is done recursively, so must the inner product.

[jebej]

I'll close this since most of the discussion is now happening on the PR.

[ettersi]

Sorry if this is a stupid question, but did this discussion reach a conlusion? I haven't found any PR where the discussion was continued.