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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)?

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I would just call W "the matrix of the inner product." If you get W from the standard inner product by a change of variables, then it will be a Gram matrix: en.wikipedia.org/wiki/Gramian_matrix –  Qiaochu Yuan Nov 29 '10 at 9:19
    
How do I form this matrix for a particular problem? I realize that I need to 'define' orthogonality through it; how do I do that? –  kevin holzer Nov 29 '10 at 10:09
1  
What kind of particular problem? –  Qiaochu Yuan Nov 29 '10 at 10:34
    
SVD is a different beastie; briefly, it's a numerically stable way of simultaneously determining the eigensystems of $\mathbf A\mathbf A^T$ and $\mathbf A^T\mathbf A$ when you are given $\mathbf A$. The singular values of $\mathbf A$ are the eigenvalues of those matrix products. –  J. M. Nov 29 '10 at 12:54
    
For constructing an appropriate inner product, it's application-dependent, as Qiaochu says. –  J. M. Nov 29 '10 at 12:55

1 Answer 1

up vote 3 down vote accepted

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.

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Thanks so much! That is really quite a clear explanation. –  kevin holzer Nov 29 '10 at 18:17
    
Also, I don't have enough rep to upvote your answer, but I can say thank you :) –  kevin holzer Nov 29 '10 at 19:00
    
Thanks, and you are welcome. –  yasmar Nov 29 '10 at 20:37

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