# Inner Product, Orthonormal basis, and Kronecker Delta

My professor says that any inner product on an orthonormal basis is equal to the dot product, but I am so confused as to what this can mean.

Let $$C = \{e_j\}_{j=1}^n$$ be an orthonormal basis of an inner product space $$V$$. Suppose an inner product is given $$g(u,v) = \left$$ where $$A:V \to V$$ is a linear operator that is symmetric and positive definite.

It would seem to me that $$g(e_j, e_k) = \left = \sum_{l=1}^n A_{lk}\left< e_j, e_l \right> = \sum_{l=1}^n A_{lk}\delta_{jl}= A_{jk}$$ which may or may not itself be equal to the kronecker delta $$\delta_{ij}$$.

In this inner product the basis vectors are not orthonormal because $$A_{ij}$$ is not a kronecker delta, but I had assumed them to be initially. This is one way of communicating what I am confused about.

• You have two inner products. the $e_i$'s are orthonormal for the old one $\langle\rangle$ but not for the new one $g.$ Mar 17 at 22:28
• What the professor means is that any inner product on a basis orthonormal with respect to that inner product is just the dot product. Mar 17 at 22:30

You have defined a new inner product. I think what was meant (as eyeballfrog mentioned in the comments), is that the inner product of the space $$V$$ is the dot product when the basis is orthonormal with respect to that inner product.
To see this we take $$v,w\in V$$ and write them out as $$v=a_1e_1+\dots+a_ne_n$$, $$w=b_1e_1+\dots+b_ne_n$$. Then by taking the inner product, we get:
$$\langle v,w\rangle=\langle\sum_{i=1}^n a_ie_i,w\rangle=\sum_{i=1}^na_i\langle e_i,w\rangle=\sum_{i=1}^na_i\langle e_i,\sum_{j=1}^nb_je_j\rangle=\sum_{i=1}^n\sum_{j=1}^na_ib_j\langle e_i,e_j\rangle=\\\sum_{i=1}^n\sum_{j=1}^na_ib_j\delta_{ij}=\sum_{i=1}^na_ib_i$$
• @HaimSchatz Well if we have a non orthonormal basis $v_1,\dots,v_n$ then we just don't know anything about $\langle v_i,v_j\rangle$ and can't really simplify the double sum which appeared in my answer. So you're just going to be stuck with the double sum. Mar 19 at 11:21