
Constrained Optimization
http://img261.imageshack.us/img261/5851/29531326.gif
(a) The reference they were referring is : "Find a lower triangular matrix L, and a diagonal matrix D such that $\displaystyle A = LDL^T$". I don't see how this is relevant though.
Here's what I've done so far
$\displaystyle L= x+y+z \lambda(x^2+y^2+z^2k)$
$\displaystyle L_x = 12 \lambda x = 0 $
$\displaystyle L_y = 12 \lambda y = 0 $
$\displaystyle L_z = 12 \lambda z = 0$
From this, critical points are $\displaystyle x=y=z = \frac{1}{2 \lambda}$
Since $\displaystyle f_{xx}=f_{yy}=f_{zz}=2\lambda$
The Hessian matrix will be
$\displaystyle H = \begin{bmatrix} 2 \lambda & 0 & 0 \\ 0& 2 \lambda & 0 \\ 0&0& 2 \lambda \end{bmatrix}$
So what else do I need to do here to solve this? (I can't find any similar examples from my textbook)
Also, can anyone show me how to prove part (b). I don't understand it...