Complex dot product doesn't conjugate

[cshaa]

Inner product on ℂ is supposed to be sesquilinear – it should conjugate one of the two vectors before multiplying them. Source.

const I = math.complex(0,1)
math.dot( [1,0], [I,0] ) // I
math.dot( [I,0], [1,0] ) // I

The current dot product clearly doesn't conjugate neither one of the vectors. This is also true of multiply(Vector, Vector) which apparently has a different implementation than dot.

While the Wikipedia article I linked uses the convention <v, w> = vᵀ w̅ (inner product is linear in the first argument and antilinear in the second argument), the opposite convention <v, w> = v⁺ w = v̅ ᵀ w is used as often, if not more. (For example in quantum mechanics, the first argument is exclusively antilinear.)

[josdejong]

Ah, that's interesting. The dot function simply uses multiply internally and doesn't do anything special for complex input values.

What would be needed to fix this behavior for complex values? Wrap the function in conj(...)?

[cshaa]

Yes, wrapping one of the vectors in conj would solve the issue. The question is:

1. Do we mind changing the API? (This is potentialy a breaking change.)
2. Which of the two vectors should be conjugated? (I strongly root for the first one.)

[josdejong]

Thanks Michal!

1. Since the behavior changes (though you could maybe pitch it as a bug fix) I think it's best to make it a major, breaking change. Which is no problem at all, it's just changing a different number in the version number :)
2. I'm not an expert on this subject, so I think it would be best to follow what other math applications do (Matlab, Mathematica, Python numpy, ...). What do you think?

[cshaa]

Making a survey on how other applications do it is always a good idea! I checked Mathematica, Matlab and two Python libraries – Numpy and Scipy. I was surprised that only two of these four tools had a sesquilinear dot product. Both of them conjugated the first vector.

Mathematica

Mathematica has a dot product which doesn't conjugate [discussion] [docs].

Matlab

Matlab conjugates the first argument [docs]

A = [1i 0];
B = [1 0];
dot(A, B)
% ans = 0.0000 - 1.0000i

Numpy

Numpy has two dot products. First one, dot, is meant for general N-dimensional arrays and it doesn't conjugate [docs]:

>>> np.dot([2j, 3j], [2j, 3j])
(-13+0j)

The second one is vdot and it performs “the dot product of two vectors”. It conjugates the first argument [docs]:

>>> a = np.array([1+2j,3+4j])
>>> b = np.array([5+6j,7+8j])
>>> np.vdot(a, b)
(70-8j)
>>> np.vdot(b, a)
(70+8j)

Sympy

Classic sympy doesn't conjugate [live]:

>>> M = Matrix([I, 0])
>>> M.dot(M)
-1

Sympy for QM conjugates symbolically, but since the operations are coordinate-independent, it's impossible to tell which of the vectors is conjugated [live]:

>>> from sympy.physics.quantum import *
>>> A = Ket('A'); B = Ket('B')
>>> conjugate( Dagger(A) * B ) == Dagger(B) * A
True

[josdejong]

Sorry for the late reply. Thanks @m93a this is a very clear and useful overview! .

So conclusions are:

1. If we conjugate we do the first argument
2. There is no standard behaviour in whether or not to conjugate. So we simply have to decide. I sort of like what Numpy does by simply offering two different implementations. I think though for mathjs it makes sense to offer just one solution which does conjugate, which will be a (small) breaking change in the current behavior. Do you agree? If so, I think Improve dot product #1773 is ready to be merged, right?

[cshaa]

Hey Jos,
Yes, I agree that in mathjs it makes sense to offer just one solution which does conjugate.
I also confirm that the PR is ready to be merged :)