What are the criteria for a change-of-basis transform to be doable in $O(n \log(n))$?

1. What is the point of change of basis?
2. Are change of basis matrices orthogonal?
3. Is orthogonality preserved in change of basis?

What is the point of change of basis?

Change of basis is a technique applied to finite-dimensional vector spaces in order to rewrite vectors in terms of a different set of basis elements. It is useful for many types of matrix computations in linear algebra and can be viewed as a type of linear transformation.

Are change of basis matrices orthogonal?

A matrix P is orthogonal if P−1=PT. A change of basis matrix P relating two orthonormal bases is an orthogonal matrix.

Is orthogonality preserved in change of basis?

so if the matrix M is orthogonal — that is, if MTM=I — then we have vT2w2=vT1w1, which shows that the dot product relative to the second basis produces the same value as the dot product relative to the first basis, and in particular that orthogonality is preserved when the basis is changed.