What can I say about matrix A, if both A and I-A are positive semidefinite?

Question

What can I say about matrix $A$, if both $A$ and $I-A$ are positive semidefinite? For example, is it a projection matrix then?

Answer 1

You can say $A$ is a hermitian matrix (or symmetric, in the real case) whose eigenvalues are all in the interval $[0,1]$.

Answer 2

You have already got some good answers, but I would like to add one more thing,

Also $\bf I-A$ will have pairwise eigenvalues $\lambda_k({\bf I-A}) = 1-\lambda_k({\bf A})$ so they will be each others spectral complementaries in the sense that $$\min_{\bf v}\left\{\bf \|Av\| + \|(I-A)(v+d)\|\right\}$$ will pull $\bf v$ in different directions (either towards $\bf 0$ or $\bf -d$) linearly weighted according to the $\lambda_k({\bf A})$s.