An algebraic generalisation of the Krankheit-operator modelling neurological disorders

Abstract 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), which 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 $$n\times n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> matrices. We show that the left and right matrix multiplication and a Hadamard multiplications can all be described as a special structured operator.