I learnt in a introductory linear algebra class that an inner product is defined as the dot product, i.e. $\langle v, w\rangle = \Sigma_i (v_i \cdot w_i)$
I recently learnt that inner product is actually a generalization of dot product, and an inner product satisfies certain conditions (e.g. as listed on MathWorld).
One inner product for $\mathbb{C}^n$ is given by
$\langle v,w\rangle = v^\dagger w$, where $v^\dagger$ is the transpose of the complex conjugate, i.e. $v^\dagger = \bar{v}^T$
Are there other inner products? I'm having a hard time seeing how something else would be useful. e.g. I can define norm using the inner product definition above, so it is useful, how would some other inner product be useful?
Best Answer
Any inner product on $\Bbb C^n$ is of the form $$ \langle v, w\rangle = v^\dagger Sw $$ for some positive-definite Hermitian matrix $S$. It's what you get when changing basis without changing the (geometric) inner product. In other words, if you want to preserve the dot product through a basis change, even though all vectors get represented by new components, then this is the new algebraic form of the same product.