Schur product for complex numbers.

[rkrsnan]

For matrices, Schur product or Hadamard product is defined as the entry wise product.
I want to know if they have a similar type of multiplication for complex numbers.
That is (a+ i b) o (c + i d) = (a c + i b d)
I encounter a situation where such a definition is useful.

In physics I get an expression that looks like the following

(Cos[x1] , Cos[x2] ).A.Transpose[(Cos[y1] , Cos[y2] )] +
i (Sin[x1] , Sin[x2] ).B.Transpose[(Sin[y1] , Sin[y2] )]
where A and B are 2x2 real matrices.

I can express the above expression in the following simpler form, if the complex product "o" as I defined earlier already exists in literature.

(exp[i x1], exp[i x2]) o (A+iB) o Transpose[(exp[i y1], exp[i y2])]

Thanks very much for the help.

 

[fresh_42]

You can define whatever you want, as long as it is well defined, but this isn't an issue here. The question is which properties your multiplication should have. E.g. you get something here which has little in common with complex multiplication, so the two will have to be strictly separated. All rules will have to be proven in advance though.