  It is known that singular values of idempotent matrices are either zero or
larger or equal to one \cite{HouC63}. We state exactly how many singular values
greater than one, equal to one, and equal to zero there are. Moreover, we
derive a singular value decomposition of idempotent matrices which reveals a
tight relationship between its left and right singular vectors. The same idea
is used to augment a discovery regarding the singular values of involutory
matrices as presented in \cite{FasH20}.
