Manipulate doesn't work for plotting a region where a matrix is positive semi-definite

Question

I'm facing the following problem and would very much like you help. Thanks is advance!

I have defined the following 4x4 matrix: \begin{equation} S=\left( \begin{array}{cccc} \frac{1}{r^2} & 1 & 1 & t\sqrt{x}\\ 1 & \frac{1}{t^2} & 1 & \frac{\sqrt{x}}{t}\\ 1 & 1 & 1 & \sqrt{q} \\ t\sqrt{x} & \frac{\sqrt{x}}{t} & \sqrt{q} & 1 \end{array} \right). \end{equation} where $(t,r)$ are two parameters (which I can choose arbitrarily to be any value in the segment $(0,1]$) and $(x,q)$ are the variables (each over the domain $[0,1]$).

Now, for a fixed pair of $(r,t)$ I want to plot a region of $(x,q)$ over which the matrix $S$ is positive semi-definite (I want it to be a valid covariance matrix). The thing is that when I try to use the Manipulate function (manipulation on $(r,t)\in(0,1]^2$), and plotting the region of $(x,t)$, nothing shows. Namely, Iv'e written the following:

S={{1/r^2, 1, 1, t Sqrt[x]}, {1, 1/t^2, 1, Sqrt[x]/t}, 
   {1, 1, 1, Sqrt[q]}, {t Sqrt[x], Sqrt[x]/t, Sqrt[q], 1}};

Manipulate[
  RegionPlot[PositiveDefiniteMatrixQ[S], 
             {x, 0, 1}, {q, 0, 1}], {r, 0, 1}, {t, 0, 1}]

and I get an empty graph no matter what values I set for $(r,t)$. You might say that maybe this matrix is never positive semi-definite, but the strange this is that if I manually input numerical values to $(r,t)$ and plot the region, everything works fine.

I really need to manipulate over all possible parameter value of $(r,t)$. What is wrong here and how can I fix it?

Moreover, is it possible for Mathematica to parametrically find a local minima of a certain function (of the variable $q$ where $(r,t,x)$ serve as parameters) subject to the constraint that the above matrix must be positive semi-definite? If so, it would very much help me to know how to formalize such a command.

Answer 1

Try to define your matrix as a function of $(x,q,r,t)$ variables

S[x_, q_, r_, t_] := {{1/r^2, 1, 1, t Sqrt[x]}, {1, 1/t^2, 1, Sqrt[x]/t}, {1, 1, 1, Sqrt[q]}, {t Sqrt[x], Sqrt[x]/t, Sqrt[q], 1}};

Then Manipulate does what you want

Manipulate[
 RegionPlot[
  PositiveDefiniteMatrixQ@S[x, q, r, t], {x, 0, 1}, {q, 0, 1}],
   {{r, 1/2}, 0, 1}, {{t, 1/2}, 0, 1}]