<s>[INST] Values that make matrix positive definite Assume the symmetric matrix:
$$
M=
  \begin{bmatrix}
    \frac{\sigma\omega\pi^2}{4L^2} + g & 0 & -\frac{\sigma + c^2Qg}{2\sigma} \\
    0 & a\mu & -\frac{Q}{2}\left(\frac{c^2}\sigma + g - \mu \right) \\
    -\frac{\sigma + c^2Qg}{2\sigma} & -\frac{Q}{2}\left(\frac{c^2}\sigma + g - \mu \right)& \frac{c^2Q}{\sigma}
  \end{bmatrix}
\
$$
with $\omega \in (0,1)$, $\sigma > 0$, $\mu > 0$.
The objective is to find some values of $Q,a > 0, g \in \mathbb{R}$ that make $M$ positive definite.
Attempt:
Using Sylvester's criterion, if the determinants:
\begin{align}
d_1 &= \left( \frac{\sigma\omega\pi^2}{4L^2} + g\right)a\mu \\
d_2 &= \left[ \frac{a\mu c^2Q}{\sigma}- \frac{Q^2}{4}\left(\frac{c^2}{\sigma} + g -\mu \right)^2\right]\left( \frac{\sigma \omega \pi^2}{4L^2} + g \right) - a\mu \left(\frac{\sigma + c^2Qg}{2\sigma} \right)^2
\end{align}
are positive then $M$ is positive definite.
$d_1$ is easy to make it positive, for $g = 0$ and $a = 1$ for example. But then, I'm having trouble finding a value for $Q$ such that:
$$
d_2 = \left[ \frac{\mu c^2Q}{\sigma}- \frac{Q^2}{4}\left(\frac{c^2}{\sigma} -\mu \right)^2\right]\frac{\sigma\omega\pi^2}{4L^2} - \frac{\mu}{4} > 0
$$
Any thoughts? Is my initial choice of $g,a$ good?
 [/INST] You may take $g=0$ and $a=Q^2$. Then $d_1$ is obviously positive and
\begin{align}
d_2
&= \left[ \frac{a\mu c^2Q}{\sigma}- \frac{Q^2}{4}\left(\frac{c^2}{\sigma} + g -\mu \right)^2\right]\left( \frac{\sigma \omega \pi^2}{4L^2} + g \right) - a\mu \left(\frac{\sigma + c^2Qg}{2\sigma} \right)^2\\
&= \left[ \frac{\mu c^2Q^3}{\sigma}- \frac{Q^2}{4}\left(\frac{c^2}{\sigma} -\mu \right)^2\right]\left( \frac{\sigma \omega \pi^2}{4L^2} \right) - Q^2\mu \left(\frac{\sigma}{2\sigma} \right)^2\\
&= \frac{\mu c^2Q^3}{\sigma}\left( \frac{\sigma \omega \pi^2}{4L^2} \right) - Q^2\times\text{constant}\\
\end{align}
is positive when $Q>0$ is sufficiently large.
</s>