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We address the existence of non-trivial closed invariant ideals for positive operators defined on Banach lattices whose order is induced by an unconditional basis. In particular, for banddiagonal positive operators such existence is characterized whenever their matrix representations meet a positiveness criteria. For more general classes of positive operators, sufficient conditions are derived proving, particularly, the sharpness of such results from the standpoint of view of the matrix representations. The whole approach is based on studying the behavior of the dynamics of infinite matrices and the localization of the non-zero entries. Finally, we generalize a theorem of Grivaux regarding the existence of non-trivial closed invariant subspaces for positive tridiagonal operators to a more general class of band-diagonal operators showing, in particular, that a large subclass of them have non-trivial ideals.