What Is the Inner Product <V,s> in Complex Vector Projections?

[polaris90]

I need some clarification on projections of complex vectors. If I have a nxm matrix of complex numbers V and a mx1 matrix s, and I want to find the projection of s onto any column of V. The formula to do this is

c = <V, s>/||V(j)||^2 where V(j) is the column of V to be used. My question is, what is <V,s>? is that the inner product of the whole matrix V with s, or is it the inner product of V(j) with s? Where V or V(j) would be the Hermition of the vector.

 

[Fredrik]

Where did you find this formula? I'm only familiar with projections of vectors onto other vectors and onto subspaces. I have no idea what <V,s> means, or what the projection of s onto V(j) means, since V(j) isn't a member of the same vector space as s.

 

[polaris90]

I see I wasn't clear on my question. What I meant is the projection of vector vector s onto a vector V. By V(j) I meant a column of matrix V. <V, s> is the inner product of V and s. I know about projections of one vector onto another vector when they are all real numbers. In this case, I have a matrix V with complex numbers. I want to project s onto a column of V. I hope my question is clearer now.

 

[Fredrik]

But if V is an n×m matrix and s is not, how can you be talking about the inner product of V and s? An inner product takes two members of a vector space (the same vector space) to a number, but V and s aren't in the same vector space. Also, if s is m×1, and V(j) is n×1, they're not in the same vector space either (unless of course n=m).

Another thing: You seem to be thinking of "vectors" as ordered sets of numbers. That's not always the case. The members of any vector space are called vectors. A vector space V is considered "complex" when the scalar multiplication operation is a function from V×ℂ into V. There are real vector spaces whose members are matrices with complex entries.