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This paper considers  N  ×  N  matrices of the form  A   γ  = A + γ  B , where  A  is self‐adjoint,  γ ∈ C  and  B  is a non‐self‐adjoint perturbation of  A . We obtain some monodromy‐type results relating the spectral behaviour of such matrices in the two asymptotic regimes | γ | → ∞ and | γ | → 0 under certain assumptions on  B . We also explain some properties of the spectrum of  A   γ   for intermediate‐sized  γ  by considering the limit  N  → ∞, concentrating on properties that have no self‐adjoint analogue. A substantial number of results extend to operators on infinite‐dimensional Hilbert spaces.

Sectorial perturbations of self‐adjoint matrices and operators