I am trying to solve this equation my self with my knowledge about characteristic polynomials, etc but I have placed it here earlier because I'm not a mathematician and maybe you give me ideas to solve the problem faster.
Consider the following matrices:
$C=\begin{bmatrix}C_{11}&C_{12}&C_{13}\\C_{12}^*&C_{22}&C_{23}\\C_{13}^*&C_{23}^*&C_{33}\end{bmatrix}$
$C_{\text{vol}}(\theta_0,\sigma^2)=[C_\alpha]+p(\sigma^2)[C_\beta(2\theta_0)]+q(\sigma^2)[C_\gamma(4\theta_0)]$
$[C_\alpha]=\frac{1}{8}\begin{bmatrix}3&0&1\\0&2&0\\1&0&3\end{bmatrix}$
$[C_\beta(2\theta)]=\frac{1}{8}\begin{bmatrix}-2\cos(2\theta)&\sqrt{2}\sin(2\theta)&0\\\sqrt{2}\sin(2\theta)&0&\sqrt{2}\sin(2\theta)\\0&\sqrt{2}\sin(2\theta)&2\cos(2\theta)\end{bmatrix}$
$[C_\gamma(4\theta_0)]=\frac{1}{8}=\begin{bmatrix}\cos(4\theta)&-\sqrt{2}\sin(4\theta)&-\cos(4\theta)\\-\sqrt{2}\sin(4\theta)&-2\cos(4\theta)&\sqrt{2}\sin(4\theta)\\-\cos(4\theta)&\sqrt{2}\sin(4\theta)&\cos(4\theta)\end{bmatrix}$
$p(\sigma^2)=\frac{2(1-\sigma^2)(1-2\sigma^2)}{1+\sigma^2}$
$q(\sigma^2)=\frac{(1-\sigma^2)(1-2\sigma^2)(1-3\sigma^2)(1-4\sigma^2)}{(1+\sigma^2)(1+2\sigma^2)(1+3\sigma^2)}$
Where $0\le\sigma^2\le 1$ and $-\pi\le\theta_0\le\pi$
Then what is the maximum value of the real parameter $f_v$ in terms of $\theta_0$ and $\sigma^2$ with the constraint that the following matrix $A$ is positive semi-definite?
$$A=C-f_vC_{\text{vol}}(\theta_0,\sigma^2)$$