Idempotent matrix of the form $(D-A)$

Question

Does there exist an idempotent matrix of the form $P=(D-A)$ where $P^2 = P$ if $A$ is idempotent? $D$ is a diagonal matrix with positive distinct entries. For the trivial case when $D$ is the identity matrix, \begin{align} (I - \frac{aa^T}{a^Ta})(I - \frac{aa^T}{a^Ta}) &= I - 2\frac{aa^T}{a^Ta}+\frac{(a^Ta)aa^T}{(a^Ta)^2}\\ &= I - \frac{aa^T}{a^Ta} \end{align} I am trying to figure this out because I have to take the square root of a matrix of a diagonal matrix plus a positive definite symmetric matrix. If I can figure out some $P$, I should be able to find a way.

Answer 1

For a positive definite $D$, $P$ only exist if $D=I$ since $I$ is the only diagonal matrix that commutes with an idempotent matrix $A$.

The sum of a diagonal matrix plus a positive definite matrix is positive definite. Thus, you can use Cholesky decomposition. If the matrix is sparse, the Incomplete Cholesky decomposition is a more efficient alternative.