Rank and determinant of an even-order tensor

Abstract

A supermatrix is a representation of a tensor with respect to a fixed basis. It can be seen a generalization of a matrix representation of a second order tensor. We define the determinant and the rank of an arbitrary even order supermatrix and show that the determinant is invariant under change of basis whose action is given by a special linear group (i.e., the determinant of the matrix representation of the basis change is 1). Moreover, we show that the rank is invariant under any change of basis.