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

Dot computes the standard dot product (a tensor contraction). It is not an inner product in the mathematical sense. This is noted in Possible Issues in the reference guide page for Dot.

Also it is by design. For one, we do not want a complicated mess emerging from symbolic dot products. More importantly, Dot is a general function that takes multiple arguments and allows for tensors of arbitrary rank (subject to dimensional compatibility requirements of neighboring arguments). The Hermitian inner product is a far more specialized beast, and what it requires does not extend to Dot.

The Scope section of documentation has at least one example of computing as a Hermitian inner product. Likewise, in the ref guide page of Inner there is an example of this.

Thank you, Daniel.

That is helpful.

If mat1 is a 1x3 complex-valued matrix, mat2 is 3x3, and mat3 is 3x1, what part(s) would be conjugated in taking the product mat1.mat2.mat3?

The fact that Dot does not do conjugation will be mentioned under "Possible Issues" in the reference page for Dot, as of the next release. Likewise there will be an example of using Inner to get the scalar product of vectors in C^n, I believe under the "Scope" section on the page for Inner.

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??

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.

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.

I see your point. Mathematica implements the dot product in the usual way, even for complex numbers. Which is not suitable as an inner product over a complex vector space. I don't know if there is a built in function for this, but you can implement your own:

complexInner[a_, b_] := Conjugate[a].b

This conjugates the first argument; you could in the same manner conjugate the second argument instead.

Best, David

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, 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