What is the name of the matrix used to weight an inner product? In Linear Algebra, when computing an inner product $<x,y> = y^*Wx$, what is the name of the matrix W?  
If it doesn't have a name, where can I find a practical explanation of how to construct it for a particular problem or space?  
Is there a text on the subject that explains this simply? A note on my background, I am approaching this from the perspective of an engineering student and not that of a mathematician; I don't have an understanding of the finer points of topology or differential geometry (not yet at least :) ).
I believe that this is the same matrix used in vector calculus to perform a change of variables. As in, we perform our change of variables, then multiply the new expression by $\frac{det(J(W))}{det(J(V))}$ where $W$ is this magic matrix in the new space, $V$ is this magic matrix the old space, and J(X) is the Jacobian operator.  Am I correct?
Also, is this related to one of the the matrices that comes up in Singular Value Decomposition (namely the diagonal matrix containing the singular values)?
 A: I think you will need to give more details on what is motivating your search for $W$ before you'll get a satisfactory answer. I will make a guess that you have a vector space that has a preferred basis and you are seeking an inner product which renders this basis orthonormal.
I don't know of a name for $W$; it is the matrix of the inner product. It should be (symmetric) positive definite.
Suppose we have $d$ linearly independent vectors $\{x_i\}$ in $\mathbb{R}^d$ that are a convenient basis for our work, except that they aren't orthonormal. We may as well assume they are normalized with respect to the usual inner product on $\mathbb{R}^d$. Make a matrix $X$ whose columns are these vectors. Then $X$ is the matrix of a change of basis. Specifically, $X^{-1}y=x$ gives us $x$ as the representation of $y$ in the basis $\{x_i\}$. 
Now consider $x^Tx$. That is the inner product we want. It is the inner product which renders our basis orthonormal. Since $x^Tx = y^TX^{-T}X^{-1}y$, we can set $W = X^{-T}X^{-1}$ and we have the desired inner product without having to explicitly perform a change of basis. (Here $X^{-T}$ is the inverse of the transpose of $X$.)
One context where I have dealt with this is when $X$ was the complete basis of eigenvectors of a linear operator, and I wanted an inner product for which the operator was self-adjoint.
