Given a matrix in a certain basis (say, canonical) and the matrix in another (orthogonal) basis $X$, is there a way to find out the basis $X$ itself?
$$A = X^T B X$$
where the matrices $A$ and $B$ are known, $X$ is unknown.
I tried to do some algebraic manipulations but I cannot keep $X$ in only one side of the equation.
[edit] Responding comments:
- The matrices $A$ and $B$ are not scalar.
- A computational approximate is sufficient.
- Even if the solution is non-unique, I need to find some $X$ such that the above relation is valid.