Berezinian

The Berezinian, also known as the superdeterminant, is the analogue of the determinant for matrices acting on Z2-graded (super) vector spaces. It plays a central role in superlinear algebra and in the integration theory of superspace, where ordinary determinants are replaced by the Berezinian in change-of-variables formulas.

Consider a supermatrix M written in 2-by-2 block form with A of size p×p, D of size q×q, and off-diagonal blocks B and C, so M = [A B; C D]. If D is invertible, the Berezinian is Ber(M) = det(A − B D^{-1} C) det(D)^{-1}. If A is invertible, an equivalent expression is Ber(M) = det(A) det(D − C A^{-1} B)^{-1}. In general, Ber is defined for invertible even endomorphisms of the graded space and is a rational function of the matrix entries; it reduces to the usual determinant in the purely bosonic case.

Key properties of the Berezinian include multiplicativity Ber(MN) = Ber(M) Ber(N) and Ber(I) = 1. For a block-diagonal matrix M = diag(A, D), Ber(M) = det(A) det(D)^{-1}, illustrating that unlike the ordinary determinant, the blocks contribute with opposite signs.

Applications include the change of variables formula in integration on superspaces. When integrating functions of both commuting (even) and anticommuting (odd) variables, the Jacobian factor that appears under a coordinate transformation is the Berezinian of the transformation matrix, ensuring the integral transform correctly under superalgebraic symmetries. The Berezinian thus generalizes the determinant to the setting of supergeometry.