Again, we have had this issue in our Math-classes. They explained us that there are two definitions of the inner product of complex vectors:

• mathematical definition: a.b = Conjugate[a].b
• Definition used by physicists: a.b=a.Conjugate[b]

BUT there is no definition for an inner product of complex vectors without a conjugate.

In my eyes, Mathematica should provide us

• with the Dot-function calculating the correct inner product with the mathematician-definition; AND
• a special function for calculating the inner product according to the physicists-definition.

Best Regals,

JJJ

Daniel @Daniel Lichtblau,

I see you added this issue in the documentation several years ago under "Possible Issues" - Thank you. The problem I see is that many engineers do not understand what a "Hermitian inner product" is. I had to look it up since that term was unfamiliar to me. Can you expand that section of the documentation a bit to also say that issue applies to complex vector math?

It would also be helpful to have a statement about this issue in the Details section that links to the Possible Issues because Dot is so frequently used for complex vector space operations.

Regards,

Neil

4 years after your initial discussion, there is still no warning flag in the reference that the Dot-product for complex vectors does not match the mathematical definition. There is one very small hint in the definition of the Norm under "Properties & Relationships". It was one full afternoon to discover that there is (in my eyes) a bug in Mathematica.

Daniel,

I am not sure that is correct. On Matlab's site they give an example of a 4 dimensional complex dot product. The Mathematica result and the Matlab result are different. I believe that Matlab is correct. I may be missing something but the first line gives the Mathematica result and the second line gives the Matlab result. I would normally expect Mathematica to test for complex arguments and apply the definition given at the bottom of the Matlab website (as Matlab does). As the alternative they should discuss this in the documentation (as you suggest) (with an example?). Thank you.

In[1]:= Dot[{1 + I, 1 - I, -1 + I, -1 - I}, {3 - 4 I, 6 - 2 I, 
  1 + 2 I, 4 + 3 I}]

Out[1]= 7 - 17 I

In[2]:= Dot[
 Conjugate[{1 + I, 1 - I, -1 + I, -1 - I}], {3 - 4 I, 6 - 2 I, 
  1 + 2 I, 4 + 3 I}]

Out[2]= 1 - 5 I

To respond to the issue of complex conjugation, Dot is doing what it was designed to do. It operates on general tensors and, more importantly, is vararg (in other words, not restricted to two arguments). The scalar product under discussion above, in contrast, has arity 2, that is, must have exactly two arguments.

To get the Hermitian inner product one can use Inner, as below.

aa = {1 + I, 3 - I, -5 + 7*I};
bb = {-2, -3 - I, 6*I};
Inner[#1*Conjugate[#2] &, aa, bb, Plus]

There is an open suggestion that this be documented better. I'll add a note referencing this Community thread.

(* Out[1705]= 32 + 34 I *)

My earlier response referred to C rather that Cn, so I have edited it out.

Some good material on complex vector spaces can be found at: Complex Vector Spaces

Hello Neil,

My source for the definition the scalar product when either vector has complex components is a text 'Mathematical Methods in the Physical Sciences', by M. L. Boas 3rd Ed 2006. I will also go back to my old Morse and Feshbach 'Methods of Theoretical Physics' and see whether or not they offer a view.

David,

I think you found a bug. I've been burned by this in the past. I just reported this to wolfram -- Either Dot should work properly for complex numbers or Wolfram should note this in the documentation and say how to extend it for Complex numbers.

In researching this I am a bit confused -- maybe you Complex math gurus can help explain this. In Wikipedia it says the Complex dot product is a.Conjugate[b] but in Matlab's website it says its Conjugate[a].b. They are not the same (They are conjugates of each other). Which is correct?

Regards

Hi David,

Thanks again for your help. It just goes to show that it always pays to start with some really simple examples that can be checked by hand. All that ancient Fortran coding experience still pays off from time to time!

Cheers,

David

The way you write vectors, they become column matrices, for which the Dot product is inappropriate. You can work around it for example by transposing or flattening:

v1 = {{5 + 2 I}, {3 - I}, {1 + I}};
v2 = {{-6 - 3 I}, {4 + I}, {5 - 7 I}};
Transpose[v1].v2
Flatten[v1].Flatten[v2]

However, in Mathematica you should really write vectors as simple lists, even if they will not display as columns.