It is known that singular values of idempotent matrices are either zero or larger or equal to one {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 {FasH20}.
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Faßbender et al. (2024) studied this question.
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