Question

Complex vectors, dot product and matrix product. Little discussion here: For the usual Euclidean dot product to be geometrically meaningful with complex vectors, the latter vector in the product \(a\cdotb\) is taken as the complex conjugate instead. What bothers me is that generally when discussing matrix multiplication, it is explained as an algorithm that consists of row and column vector dot products. In the case of matrices having complex number entries, why should the complex conjugate version of the dot product be ditched? Or are we just speaking figuratively...

Answer 1

What dimension we talking about here?

Answer 2

And why do u think complex vectors should ever be geometrically meaningful?

Answer 3

Whenever the matrix multiplication is defined. For instance, for matrices \(A=[a_{ij}]\) and \(B=[b_{ij}]\) with orders \(r\times p\) and \(p \times c\), respectively, why is the matrix multiplication defined by \(c_{ij}=\sum\limits_{k=1}^p a_{ik}b_{kj}\) and not \(c_{ij}=\sum\limits_{k=1}^p a_{ik}\bar b_{kj}\)?

Answer 4

Of course it doesn't necessarily need to be geometrically meaningful, but I'd prefer salvaging for example the property that you won't get zero if neither of the two vectors are zero vectors.

Answer 5

if you defined multiplication using the conjugate then you will lose some properties...for example the determinant of a product would not be the product of the determinants

Answer 6

Also I think u have not to mix up implementation (in a program/algorithm) with what happens in the algebra.

And in the algebra, there are different inner products possible not just dot (although in R3 it would typically be dot).