Several neurological disorders can be described as alterations of the brain connectome, both anatomic and functional. To model diseases and compare them, it has been proposed the Krankheit-Operator (K-operator), that acts on the weights of the connectome, reproducing the effects of specific disorders. In this article, with algebraic tools, we attempt to provide a more general definition of the operator, that encompasses the previous different definitions provided.  We consider a general setting where the linear operator is an endomorphism on the vector space of nxn matrices. We show that the left and right matrix multiplication and a Hadamard multiplications can all be described as a special structured operator.