  It is a well-known result of T.\,Kato that given a continuous path of square
matrices of a fixed dimension, the eigenvalues of the path can be chosen
continuously. In this paper, we give an infinite-dimensional analogue of this
result, which naturally arises in the context of the so-called unitary spectral
flow. This provides a new approach to spectral flow, which seems to be missing
from the literature. It is the purpose of the present paper to fill in this
gap.
