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}.

翻译：已知幂等矩阵的奇异值要么为零，要么大于或等于一\cite{HouC63}。我们精确刻画了其中大于一、等于一及等于零的奇异值数量分布。进而，我们推导出幂等矩阵的奇异值分解，揭示了其左、右奇异向量之间的紧密关联。基于相同思路，我们补充了\cite{FasH20}中关于对合矩阵奇异值的结果。