dot product for complex vector

Hello,

In the Matlab example, you have the dot product of the following two vectors A and B and its answer is vector C.

A = [1+i 1-i -1+i -1-i];

B = [3-4i 6-2i 1+2i 4+3i];

Calculate the dot product of A and B.

C = dot(A,B)

C = 1.0000 - 5.0000i

However, when I calculate it, I have vector C = 7 - 17i

That is, I have C vector results as follows below

(1+i) * (3-4i) + (1-i) * (6-2i) + (-1+i) * (1+2i) + (-1-i) * (4+3i) =

(7-i) +( 4-8i) + (-3-i) + (-1-7i) =

7 - 17i.

Hence, could you please tell me how the Matlab got the results (or show me manually how Matlab got the dot product answer) as I have different results than Matlab calculated using dot product?

Thank you,

Charles

Answer 1

[John D'Errico]

help dot
 dot  Vector dot product.
    C = dot(A,B) returns the scalar product of the vectors A and B.
    A and B must be vectors of the same length.  When A and B are both
    column vectors, dot(A,B) is the same as A'*B.
    
 dot(A,B), for N-D arrays A and B, returns the scalar product
    along the first non-singleton dimension of A and B. A and B must
    have the same size.

So what are A and B?

A = [1+i 1-i -1+i -1-i];
B = [3-4i 6-2i 1+2i 4+3i];

They are row vectors, complex row vectors. So MATLAB forms the result as:

dot(A,B)
ans =
            1 -          5i

sum(conj(A).*B)
ans =
            1 -          5i

That is, when A and B are both vectors, MATLAB treats them the same as if A and B were column vectors. It effectively thinks of them as both column vectors. Then it forms the result by conjugating A, takes an element-wise product, then sums those terms.