$$\mathbf{x}^TA\mathbf{x}=\sum^n_{i=1}\sum^n_{j=1}a_{ij}x_ix_j\quad \mathbf{x}^T=(x_1,x_2,..x_n)$$
I know above equality holds when it's in the concrete examples, I can justify it but when it becomes abstract above form, I hardly to understand it. Or I should ask how does one deduce from left to right. I was trying to split two sum forms to understand, but it's not really worked out.
Please help. Thank you for any comment
$$ x^T A x = x \cdot (A x)\\ (Ax)_i = \sum_j a_{ij} x_j\\ x \cdot (Ax) = \sum_i x_i (Ax)_i\\ = \sum_i x_i \sum_j a_{ij} x_j\\ = \sum_i \sum_j x_i a_{ij} x_j\\ = \sum_i \sum_j a_{ij} x_i x_j\\ $$
The steps here are rewriting $x^T y = x \cdot y$ with $y \equiv Ax$, the definition of the $i$'th entry in a matrix vector product, plugging in, distributing $x_i$ and the sum over $j$, and then rearranging scalar factors.