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.

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