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A remark on the mathematics of the seesaw mechanism

To demonstrate that matrices of seesaw type lead to a hieararchy in the neutrino masses, i.e. that there is a large gap in the singular spectrum of these matrices, one generally uses an approximate block-diagonalization procedure. In this note we show that no approximation is required to prove this gap property if the Courant-Fisher-Weyl theorem is used instead. This simple observation might not be original, however it does not seem to show up in the literature. We also sketch the proof of additional inequalities for the singular values of matrices of seesaw type.