Factorization of real square matrix into eigenvalues and symmetric and idempotent matrix

Question

Is there a theorem that allows one to decompose any real $(n \times n)$ matrix $A$ into $P \Lambda P^{\mathsf{T}}$ where $P$ is symmetric and idempotent and $\Lambda$ is a diagonal matrix of eigenvalues of $A$? If so, where can I find its proof?

Answer 1

No, there is no such theorem. It is impossible to do so.

In particular: take $A$ is invertible and non-diagonal. $P$ must be invertible in order for the product $P\Lambda P^T$ to be invertible. However, the only invertible idempotent matrix is $I$. So, we would need to have $A = I\Lambda I^T = \Lambda$. But $\Lambda$ is diagonal, and $A$ is not.

So, there is no such $P$ and $\Lambda$ for an invertible, non-diagonal $A$.