coefficients in square of sum I have the following expression:
$Ax^2 + By^2+Cz^2 + AB2xy + AC2xz + BC2yz$
I am trying to find out if the above expression is always positive.
$A,B,C$ are always positive and $x,y,z$ are always non zero
The expression looks very similar to a square of sum: $(x+y+z)^2$ but I don't know if I can factor out the $A,B,C$ coefficients.   
 A: Nope.
Let $A=B=C=2$, $x=1,y=z=-1$
The expression $=-2 < 0$
A: No, the answer would always not be positive.One example is A=4,B=1,C=1,x=-1,y=1,z=1.
A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
Hint:
$$
\pars{x \quad y\quad z}
\pars{\begin{array}{ccc}
\ds{A} & \ds{AB} & \ds{AC}
\\
\ds{AB} & \ds{B} & \ds{BC}
\\
\ds{AC} & \ds{BC} & \ds{C}
\end{array}}
\pars{\substack{\ds{x} \\[3mm] \ds{y} \\[3mm] \ds{}z}}
$$

Study the above matrix eigenvalues. What conditions must satisfy $\ds{A}$, $\ds{B}$ and $\ds{C}$ such that all eigenvalues $\ds{> 0 }$ ?.

