I want to find a matrix $M$ satisfying $$M^TPM=D$$ where
- $P$ is a known symmetric square matrix.
- $D$ is a unknown diagonal matrix.
This problem is a matrix congruence problem. We can think of this problem as an eigenvalue problem, but in order to do so, we must additionally consider the assumption that $M$ is orthonormal. However, such assumptions can not be used in this problem.
This is a congruence problem of symmetric matrix and I hope someone can help me.