Hi David, Throughout your notebook you are inclosing each complex number in its own bracket. This turns your vectors into matrices. Here is how to take the dot product of vectors:

In[1]:= {1 + 2 I}.{2 + 3 I}

Out[1]= -4 + 7 I

David, Gianluca,

Thank you both for pointing out my list structure error. My Notebook now executes without error messages, but I am not getting the results that I expected. In my new attachment, the Mathematica Dot function gives one result whereas the 'longhand' version gives the answer that I derive by hand. I am taking my definition of the vector dot product for (3D) complex component vectors to be:

A.B = Sum (from i=1,3) A*[i]B[i] as in my longhand version.

It maybe that I am using the wrong definition for the Dot product when the components are complex. I shall dig deeper tomorrow.

Again, thank you for your prompt replies.

Attachments:

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

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.

Perhaps theoretical physicists use a different definition than what I was taught in linear algebra classes. However, the two definitions are just conjugate of each other. As for Mathematica's Dot function, the documentation should address this issue, because it leads to confusion. The Dot function does tensor index contraction without introducing any conjugation. There is no built-in function for the Hermitian inner product of complex vectors. The Norm function does what we would expect in the complex case too, but using Abs, not Conjugate.

Tech Support confirmed the issue. They said they were filing "a report with our developers for them to generalize the Dot function to be able to handle complex vectors".

I don't know which version they plan on implementing -- It seems most people agree that it should be Conjugate[a].b (Like Matlab). Is the Wikipedia article simply wrong? Maybe I should just edit it and fix it??

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

Thank you, Daniel.

That is helpful.

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

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

Example:

p = {a, I b};
q = {c, I d};

In[3]:= p . q
Out[3]= a c - b d

Another:

z = {I 3, I 4};
u = {I 5, I 6};

In[6]:= z . u
Out[6]= -39