Given a matrix pseudodifferential operator on a smooth manifold, one may be
interested in diagonalising it by choosing eigenvectors of its principal symbol
in a smooth manner. We show that diagonalisation is not always possible, on the
whole cotangent bundle or even in a single fibre. We identify global and local
topological obstructions to diagonalisation and examine physically meaningful
examples demonstrating that all possible scenarios can occur.
