Inconsistent inner product with complex-valued tuples

[ocramz]

https://github.com/conal/vector-space/blob/master/src/Data/VectorSpace.hs#L145 defines (u,v) <.> (u',v') = (u <.> u') ^+^ (v <.> v'), but (1, j) <.> (j, 1) should be 2 j, rather than 0.

[leftaroundabout]

Basically the question is: why is Scalar (Complex r) ~ r, instead of Scalar (Complex r) ~ Complex r?

In fact, when the complex numbers are considered as a 2D real space, then the current result does make sense (you can't apply linearity i · 〈v,w〉 = 〈i·v,w〉 anymore – (1,i) and (i,1) are simply orthogonal then); but that would basically mean that we don't have any complex vector spaces – only real spaces, some of which are called Complex.

That's quite silly if you ask me. It certainly precludes many relevant applications, from DSP to quantum mechanics. Hence I'd propose to change the instance to

instance VectorSpace v => VectorSpace (Complex v) where
  type Scalar (Complex v) = Complex (Scalar v)
  (x:+y) *^ (x':+y') =  (x*^x' ^-^ y*^y') :+ (x*^y' ^+^ y*^x')

Of course one might now object that magnitudeSq v does not have a real number type, but we already have much the same issue with the (also not uncontroversial) function instance, whose scalars are functions, not real numbers.

This could perhaps be adressed with a separate class

class VectorSpace v => Normed v where
  type Magnitude v :: *
  magnitudeSq :: v -> Magnitude v

[ocramz]

That was exactly my conclusion, we do need a separate associated typefamily, independent of the Scalar one introduced by VectorSpace. Indeed, the norms of vectors over either R or C must be real, but the inner product of two vectors over C need not be.

I'd change the definition of Normed to reflect the fact that the L2 norm is induced by the inner product:

class InnerSpace v => Normed v where
  type Magnitude v :: *
  magnitudeSq :: v -> Magnitude v
  magnitudeSq x = x <.> x   

[ocramz]

^ no, the default definition of magnitudeSq cannot use <.> because Couldn't match expected type 'Magnitude v' with actual type 'Scalar v'.

Edit: (that Magnitude v shouldn't always unify with Scalar v is exactly the point, my bad)