  Let A and B be normal matrices with coefficients that are continuous
complex-valued functions on a topological space X that has the homotopy type of
a CW complex, and suppose these matrices have the same distinct eigenvalues at
each point of X. We use obstruction theory to establish a necessary and
sufficient condition for A and B to be unitarily equivalent. We also determine
bounds on the number of possible unitary equivalence classes in terms of
cohomological invariants of X.
