Why is inverse of orthogonal matrix is its transpose?

So the question is in the title. It's easy to prove when we know that there are real numbers in it and the dot product is standard. But why this works in the general case - when there are complex numbers inside and the dot product is something else?

Solution 1:

Let $C_i$ the $i^{\text{th}}$ column of the orthogonal matrix $O$ then we have

$$\langle C_i,C_j\rangle=\delta_{ij}$$ and we have $$O^T=(C_1\;\cdots\; C_n)^T=(C_1^T\;\cdots\; C_n^T)$$ so we get

$$O^TO=(\langle C_i,C_j\rangle)_{1\le i,j\le n}=I_n$$

Solution 2:

A is othogonal means A'A = I. That says that A' is the inverse of A!