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.

 

[blue_raver22]

The notation <V,s> typically refers to the inner product of the entire matrix V with s. This means that you would take the conjugate transpose of V (often denoted as V*) and multiply it by s. The result would be a scalar value, which is then divided by the norm squared of the column of V that is being used (V(j)).

In this case, the inner product of V(j) with s would be denoted as <V(j),s>. It is important to note that both <V,s> and <V(j),s> will result in the same scalar value, but the former refers to the inner product of the entire matrix while the latter refers to the inner product of a specific column.

Additionally, the Hermition (conjugate transpose) of a vector is used in complex vector spaces to account for the complex conjugation of the elements. This is necessary in order to accurately calculate the inner product in complex vector spaces.

I hope this clarifies your doubts regarding the notation and calculations involved in finding the projection of a complex vector onto a column of a matrix.