Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Given a basis $B = \{g_0, g_1,\ldots , g_{nk}\} $, I wish to construct a set of $nk$ functions $S$ such that $\langle S_i,S_j \rangle = \delta_{i,j}$ (so that $S$ is an orthonormal set) where $\langle f,g\rangle = \int_{0}^{\infty}f g$.

Assuming the following definitions: $$S_k = c_{k,0}\cdot g_0 + c_{k,1}\cdot g_1 +\cdots + c_{k,nk}\cdot g_{nk}$$

$C$ is a matrix where $C_{i,j} = c_{i,j}$

$G$ is a matrix where $G_{i,j} = \langle g_i,g_j \rangle $

Then it is easy to see that $CGC^T$ is a matrix with elements equal to $\langle S_i,S_j \rangle $. Thus, if $CGC^T = I$, then $C$ is the matrix containing the correct coefficients to produce the set of functions $S$. Now, upon diagonalizing the matrix $G$, we can generate a matrix $P$ where $PGP^T$ is diagonal with eigenvalues on the main diagonal.

What changes do I need to make to $P$ in order to produce $C$? Is it as simple as making small changes to $P$ or will I need to pursue another method altogether?

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

Let the diagonal elements of $PGP^T$ be $d_k$ and $P=(p_{i,j})$. Then the functions $$\tilde S_k=\sum_j p_{k,j} g_j$$ are almost what you need except that they are orthogonal but not orthonormal. The squared norms of $\tilde S_k$ are equal to $d_k$, so $$S_k=\frac{1}{\sqrt d_k}\tilde S_k.$$ As far as I see this corresponds to multiplying $k$-th row of $P$ with $\frac{1}{\sqrt d_k}$ for all $k$ to get $C$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.