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Given a non-hermitean matrix M, the structure of its minimal polynomialencodes whether M is diagonalizable or not. This note will explain how todetermine the minimal polynomial of a matrix without going through itscharacteristic polynomial. The approach is applied to a quantum mechanicalparticle moving in a square well under the influence of a piece-wise constantPT-symmetric potential. Upon discretizing the configuration space, the systemis decribed by a matrix of dimension three. It turns out not to bediagonalizable for a critical strength of the interaction, also indicated bythe transition of two real into a pair of complex energy eigenvalues. Thesystems develops a three-fold degenerate eigenvalue, and two of the threeeigenfunctions disappear at this exceptional point, giving a difference betweenthe algebraic and geometric multiplicity of the eigenvalue equal to two.Comment: 5 page

How to test for diagonalizability: the discretized PT-invariant square-well potential