Another look at the eigenvalues of functions of a pair of orthogonal projectors

The paper is concerned with eigenvalues of functions of a pair of orthogonal projectors, i.e., Hermitian idempotent matrices. By utilizing an approach based on a joint decomposition of the pair emerging from the spectral theorem, further insight into the topic is provided, supplementing the results already available in the literature. The research reveals several new facts, leading to the conclusion that the approach exploited offers a handy tool to cope with problems which require knowledge of eigenvalues of various functions of orthogonal projectors. Among the results established are characterizations of numbers of different eigenvalues of selected functions expressed in terms of ranks of the matrices involved in the joint decomposition. Related results concerned with square roots of functions determined by a pair of orthogonal projectors are provided as well.