complex bilinear extension

Hello,

I came across the following statement:

"..., where g is the complex bilinear extension of the standard scalar product on R^n to C^n (note that this a symmetric form, not a Hermitian one)."

Can someone tell me how the "complex bilinear extension" is defined, i.e. if x,y in C^n, then g(x,y)=... .

Thanks in advance!

Re: complex bilinear extension

The "standard" scalar product on R^n is where x and y are vectors in R^n. If x and y are vectors in C^n, the scalar product is where "y*" indicates the complex conjugate of y: if then . That is, if and then the inner product of x and y is .

Re: complex bilinear extension

I know that definition, but that's not symmetric or bilinear, so I don't think thats what's meant by "complex bilinear extension", which also should be symmetric.

Maybe it's what you told me, just without the conjugate.