Block diagonalisation of Hermitian matrices

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Abstract
Block diagonalisation of the Hamiltonian by an unitary transformation is an important theoretical tool, e.g., for deriving the effective Hamiltonian of the quasidegenerate perturbation theory or for determining diabatic molecular electronic states. There are infinitely many different unitary transformations which bring a given Hermitian matrix into block diagonal form. It is, therefore, important to investigate under which conditions the transformation becomes unique. The explicit construction of such a transformation and its properties is discussed in detail. An illustrative example is presented. The non-Hermitian case is briefly discussed as well.