In mathematics and theoretical physics, the Berezinian or superdeterminant is a generalization of the determinant to the case of supermatrices.
The Berezinian is uniquely determined by two defining properties: where str(X) denotes the supertrace of X.
The simplest case to consider is the Berezinian of a supermatrix with entries in a field K. Such supermatrices represent linear transformations of a super vector space over K. A particular even supermatrix is a block matrix of the form Such a matrix is invertible if and only if both A and D are invertible matrices over K. The Berezinian of X is given by For a motivation of the negative exponent see the substitution formula in the odd case.
In this case the Berezinian is given by or, equivalently, by These formulas are well-defined since we are only taking determinants of matrices whose entries are in the commutative ring R0.
In this case, invertibility of X is equivalent to the invertibility of JX, where Then the Berezinian of X is defined as The determinant of an endomorphism of a free module M can be defined as the induced action on the 1-dimensional highest exterior power of M. In the supersymmetric case there is no highest exterior power, but there is a still a similar definition of the Berezinian as follows.