Alternate inner products on Euclidean space?

Question

After reading about inner products as a generalization of the dot product, I was hoping to be able to prove that the dot product is in some sense the unique inner product in Euclidean space (e.g., up to constant scaling).

But it seems that there are a whole bunch of alternative inner products in $\mathbb{R}^2$ with nonzero cross-terms between basis vectors, for example, $\langle (a, b)^\intercal, (x, y)^\intercal \rangle = ax + by + 0.5(ay + bx)$. Unless I've made a mistake, this satisfies symmetry, linearity, and positive-definiteness.

Is there a sense in which the dot product is the canonical inner product on Euclidean space? Or do we just pick it because the implied norm matches our notion of distance?

Answer 1

Any inner product is dot product in some basis. For example, your inner product is standard dot product written in basis $\left(e_1, \frac{1}{2}e_1 + \frac{\sqrt{3}}{2}e_2\right)$.

Answer 2

There is nothing special about the dot product. Yes, it corresponds to the Euclidean norm if you are using an orthonormal basis. But if your basis is not orthonormal then the Euclidean norm will be represented by some other symmetric matrix.

Answer 3

For an arbitrary inner product $\left<\right>$ on $\mathbb R^n$, there is a positive definite real symmetric matrix $A_{ij} = \left<e_i|e_j\right>$ that defines the transform. Since it is real and symmetric, it is orthogonally diagonalizable. That is, for any inner product on $\mathbb R^n$, there is a set of real numbers $\lambda_j$ and an orthonormal basis $\left|\xi_j\right>$ such that $$ \left<a|b\right> = \sum_j\lambda_j\left<a|\xi_j\right>\left<\xi_j|b\right> $$ Roughly speaking, the inner product resolves $a$ and $b$ into their $\xi_j$ components, then weights the resulting dot product by $\lambda_j$.

In general, this choice of $\left|\xi_j\right>$ will be unique. However, for some inner products, there will be multiple possible choices of $\left|\xi_j\right>$. The Euclidean norm is unique (up to a constant scaling) in that every choice of $\left|\xi_j\right>$ allows the inner product to be written in that form--it is independent of the chosen basis.