= Solution
The terminal wealth of portfolio $\psi$ is the scalar <normal distribution> $\psi\cdot X$, with mean $\psi\cdot\mu$ and <variance> $\psi\cdot V\psi$. Its Gaussian exponential moment gives
$$
E[U(\psi\cdot X)]=-\exp\left(-\gamma\psi\cdot\mu+\frac{\gamma^2}{2}\psi\cdot V\psi\right)
=-e^{-\gamma q(\psi)}.
$$
Since $-e^{-\gamma q}$ is strictly increasing in $q$, \b[strict preference is exactly $q(\psi)>q(\theta)$]. The nonsingular <covariance matrix> is positive definite. Completing the square gives
$$
q(\psi)=\frac1{2\gamma}\mu^TV^{-1}\mu
-\frac\gamma2\left(\psi-\gamma^{-1}V^{-1}\mu\right)^TV\left(\psi-\gamma^{-1}V^{-1}\mu\right),
$$
so the unique optimum is $\boxed{\psi^*=\gamma^{-1}V^{-1}\mu}$.
Paying the deterministic transaction cost reduces the certainty equivalent of the new portfolio by $\epsilon\|z\|_1$. Put $g=\mu-\gamma V\theta$. Its gain over the current portfolio is exactly
$$
q(\theta+z)-q(\theta)-\epsilon\|z\|_1
=g\cdot z-\frac\gamma2z^TVz-\epsilon\|z\|_1.
$$
If every $|g_i|\le\epsilon$, then $g\cdot z\le\epsilon\|z\|_1$, so every nonzero trade has strictly negative gain, by positive definiteness. Conversely, if $|g_i|>\epsilon$, choose $z=t\operatorname{sign}(g_i)e_i$. Its gain is
$$
t(|g_i|-\epsilon)-\frac\gamma2t^2V_{ii}>0
$$
for $0<t<2(|g_i|-\epsilon)/(\gamma V_{ii})$. Therefore
$$
\boxed{\text{No beneficial trade exists}\quad\Longleftrightarrow\quad|\mu_i-\gamma(V\theta)_i|\le\epsilon\text{ for all }i.}
$$
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