How would we find the standard norm/inner product for non-orthonormal basis?
Good question. Here's one way to do it: suppose that the matrix A expresses the change in basis from the old orthonormal basis to the new non-orthonormal basis. In other words, if x is a vector in the old basis and x' is the corresponding vector in the new basis, then Ax = x', or equivalently, x = A^-1 x'. Hence,
< x, y > = x^T y = (A^-1 x')^T (A^-1 y') = x'^T (A^-1)^T A^-1 y.
In other words, in the new basis, the original inner product in the new basis is represented by the matrix
Thank you Prof. Keenan.
Shouldn't it be (A^-1)^T (A^-1) in the last line?
Yes, absolutely. Fixed! :-)