How to solve quadratic matrix equations of the form $A^T B A=C$? I want to solve the following matrix equation
$$A^T  
\begin{pmatrix}
 10 & 0 & 0 \\
 0 & 20 & 0 \\
 0 & 0 & 25 \\
\end{pmatrix} A =
\begin{pmatrix}
 \frac{35}{2} & \frac{5 \sqrt{3}}{2} & 0 \\
 \frac{5 \sqrt{3}}{2} & \frac{25}{2} & 0 \\
 0 & 0 & 25 \\
\end{pmatrix}$$
In the above formula, $A^T$ is the transpose of matrix A.
At present, I don't have a good way. I only know that the reference answer of matrix A is 
$$\left(\begin{array}{ccc}
 \frac{1}{2} & -\frac{\sqrt{3}}{2} & 0 \\
 \frac{\sqrt{3}}{2} & \frac{1}{2} & 0 \\
 0 & 0 & 1 \end{array}\right)$$

The matrix $C$ is real symmetric so that you can diagonalize it as
$$C=P^TDP$$ where $P$ is orthogonal (hence its transpose is its inverse).
If $D$ matches the given $B$, you have the solution. If it does not, there is no solution, as the Eigenvalues are uniquely defined.
