Why is
$$x^TAx= \sum_{j}^{n}\sum_{i}^{n} a_{ij}x_ix_j $$
$x$ is n × 1, $A$ is n × n.
What I have tried?
If $y=Ax$, then
$$y_j =\sum_{j}^na_{ij}x_{j}$$
Now, $$x^TAx= \sum_{i}^n x_iy_i $$
which becomes $$\sum_{i}^n x_i\sum_{j}^na_{ij}x_{j}=\sum_{i}^n \sum_{j}^na_{ij}x_{i}x_j$$
Now, the orders of i and j are reversed which is the problem, and confuses me.
Best Answer
$$\large{x=\begin{bmatrix} \\\end{bmatrix}_{n\times1}\to \\ x^TAx=\begin{bmatrix} \\\end{bmatrix}_{1\times n}\begin{bmatrix} \\\end{bmatrix}_{n\times n}\begin{bmatrix} \\\end{bmatrix}_{n\times1}=\begin{bmatrix} \\\end{bmatrix}_{1\times1}\to \\ \begin{bmatrix}x_1 & x_2 & ...&x_n \\\end{bmatrix}_{n\times1} \begin{bmatrix}a_{11} & a_{12} & ...&a_{1n}\\a_{21} &a_{22} &...&a_{2n}\\...\\a_{n1} &a_{n2}&...&a{nn} \\\end{bmatrix}_{n\times1} \begin{bmatrix}x_1 \\ x_2 \\ .\\.\\.\\x_n \\\end{bmatrix}_{n\times1}}$$ $$\begin{bmatrix}x_1a_{11}+x_2a_{21}+...+x_na_{n1} & x_1a_{12}+x_2a_{22}+...+x_na_{n2}& ...& x_1a_{1n}+x_2a_{2n}+...+x_na_{nn} \\\end{bmatrix}_{n\times1}\begin{bmatrix}x_1 \\ x_2 \\ .\\.\\.\\x_n \\\end{bmatrix}_{n\times1}=\\$$ $$=(x_1a_{11}+x_2a_{21}+...+x_na_{n1})x_1\\+(x_1a_{12}+x_2a_{22}+...+x_na_{n2})x_2\\+...\\+ (x_1a_{1n}+x_2a_{2n}+...+x_na_{nn})x_n\\=(\sum_{i=1}^nx_ia_{i1})x_1+(\sum_{i=1}^nx_ia_{i2})x_2+...+(\sum_{i=1}^nx_ia_{in})x_n=\\ \sum_{j=1}^n\sum_{i=1}^nx_ia_{ij}x_j $$