Orthogonal basis with respect to an inner product

Question

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

$$u\cdot v=0$$

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.

$$\langle u,v\rangle_A=0$$

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 $A$?

Answer 1

I am assuming that $\langle u,v\rangle_A=u^TAv$ for some symmetric matrix $A$ such that $(u,v)\mapsto\langle u,v\rangle_A=u^TAv$ is an inner product. Then, yes, it is called an orthonormal basis (not just orthogonal, since you are requiring that the vectors are unit vectors).

If we work with that inner product, then we will have a concept of angles, which is distinct from the usual one. But, yes, distinct vectors will be at right angles for that way of measuring angles.

Answer 2

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 u⋅v=0 for any two basis vectors u and v.

That's simply NOT true. Most vector spaces (such as the space of functions continuous on [a, b] do not have a "dot product" defined but rather a general "inner product" (for the space of functions continuous on [a, b] the inner product of f and g is $\int_a^b f(x)g(x)dx$)