Topological obstructions to the diagonalisation of pseudodifferential systems

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