
Quadratic form
let $\displaystyle x=(x_{1},x_{2},...,x_{n})$
$\displaystyle \bar{x}=\frac{1}{n}(x_{1}+x_{2}+...+x_{n})$
$\displaystyle S{_{x}}^{2}=\frac{1}{n1}((x_{1}\bar{x})^2+(x_{2}\bar{x})^2+...+(x_{n}\bar{x})^2)$
Express the quadratic form $\displaystyle S{_{x}}^{2}$ in the matrix notation $\displaystyle x^{T}Ax$, where A is symmetric.
How do i do this? its getting ugly if i only try to expand $\displaystyle S{_{x}}^{2}$
so i tried it for n=2 to get some hint of what the matrix A would be. Well i got the matrix for n=2 but i can't figure out how it would be for an arbitary n.
this is what i got for n=2 : A = $\displaystyle \begin{bmatrix}\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}\end{bmatrix}$
Thanks!