Traditionally an orthogonal basis or orthonormal basis is a basis such that all the basis vectors are unit vectors and orthogonal to each other, i.e. the dot product is $0$ or
for any two basis vectors $u$ and $v$. What if we find a basis where the inner product of any two vectors is 0 with respect to some $A$, i.e.
Is there a special name for this kind of basis or is it also just called an orthogonal basis?
Furthermore, is there a geometric interpretation for this kind of basis? If we consider the dot product, each pair of basis vectors is at right angles to each other. How does this appear geometrically if we have some general matrix $A$?