I am following this lecture notes (page 10), Using eigenvectors to find the maximum and minimum values. I am having a difficult time understanding how matrix A was derived.
Find the maximum and minimum values of
$Q(X) = x^2 + y^2 + 2z^2 -2xy +4xz+4yz$
subject to the constraint
$x^2 +y^2 +z^2 = 1 $
Matrix of Q is $$ A= \begin{bmatrix} 1 & -1 & 2 \\ -1 & 1 & 2 \\ 2 & 2 & 2 \end{bmatrix} $$
Using the "twos in" notation for quadratic forms, we have by matrix multiplication that
$$ Q(X) = X^TAX = \begin{pmatrix} x &y &z\end{pmatrix} \begin{pmatrix} a_{00} & a_{10} & a_{20}\\ a_{01} & a_{11} & a_{21}\\ a_{02} & a_{12} & a_{22} \end{pmatrix} \begin{pmatrix} x\\ y\\ z\end{pmatrix}\\ =a_{00}x^2+a_{11}y^2+a_{22}z^2+2a_{10}xy+2a_{20}xz+2a_{12}yz $$
assuming that $A$ is symmetric (i.e. that $a_{ij}=a_{ji}$). So for you to construct your matrix, you take the coefficients of the three square terms, 1,1,2, and put them on the diagonal. And take the coefficients of the three cross-terms, -2,4, and 4, divide them by 2, and put them in the appropriate off diagonal spots, giving you the desired result.