Minimizing a quadratic
Write the function
$$
f(x,y,z) = x^2 + 2xy+3y^2 +2yz+z^2 - 2x+3z+2
$$
in the form $\vec{x}^T K \vec{x} -2 \vec{x}^T \vec{f} +c$, where $K$ is a symmetric, positive definite matrix. Identify the linear system that you would then solve to find the minimizer of $f$. You do not need to solve the system.
This is essentially part of problem 5.2.1 from our text
Comments
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In order to get p(x,y,z) in the desired form, we can create a matrix K by
$$ K = \begin{pmatrix}
x^2 & xy/2 & xz/2 \\
xy/2 & y^2 & yz/2 \\
xz/2 & yz/2 & z^2 \\
\end{pmatrix}$$
which when analyzing the function and plugging in produces a K matrix
$$ K =\begin{pmatrix}
1 & 1 & 0 \\
1 & 3 & 1 \\
0 & 1 & 1 \\
\end{pmatrix}
$$
It just so happens,
$$ K^T = K $$ and $$\vec{x}^T K \vec{x} > 0$$
satisfying both the requirements for K to a positive definite matrix, which implies the minimum occurs at $$ K \vec{x} = \vec{f} $$
We look back to our polynomial and focus on $$ -2x + 3z $$
From this we can say $$ \vec{f} $$ from $$ -2\vec{x}^T \vec{f} = $$
$$ \vec{f} =\begin{pmatrix}
1\\
0 \\
\frac{3}{2} \\
\end{pmatrix}
$$
We would then need to find the x vector that satisfies Kx=f.