If a have a matrix say $A$ that is orthogonally diagonalizable (i.e. it can be written as $\lambda_1u_1u_1^T+ \lambda_2 u_2u_2^T+\dotsc \lambda_nu_nu_n^T$ , where the $u_i$ are the eigenvectors of the matrix $A$ and $\lambda_i$ are the eigenvalues).
I am just wondering is $A$ a projection matrix?
By definition, a matrix say $B$ is a projection matrix iff $B^2=B$
But it seems $A^2$ not necessarily equal $A$.
Is it correct that $A$ which is orthogonally diagonalizable not necessarily a projection matrix itself?