supermatrices

In mathematics, a supermatrix is a matrix endowed with a Z2 grading, used in the study of superalgebras and models of supersymmetry. A common presentation is to regard a supermatrix as an operator on a graded vector space, with its entries drawn from a superalgebra.

A standard format for a supermatrix of size m|n is a 2-by-2 block matrix

[A B; C D],

where A is m-by-m, D is n-by-n, B is m-by-n, and C is n-by-m. In the usual convention, the diagonal blocks A and D contain even elements, while the off-diagonal blocks B and C contain odd elements. The matrix thus preserves the grading of the underlying graded space, and such matrices form the general linear Lie superalgebra gl(m|n).

Operations on supermatrices follow ordinary matrix operations on the underlying entries, while respecting the grading. Addition and multiplication are defined entrywise, with the interpretation of the entries as elements of a superalgebra. The concept of trace extends to the supertrace, defined for a supermatrix M = [A B; C D] as str(M) = tr(A) − tr(D). The supertrace is compatible with the graded structure and plays a role in character theory of Lie superalgebras.

Inversion and determinant analogues lead to the Berezinian, the superdeterminant. For an invertible block matrix with A and D square, the Berezinian is often written as Ber(M) = det(A) det(D − C A^{-1} B)^{-1} when A is invertible (and there is a symmetric expression when D is invertible). The Berezinian generalizes the usual determinant to the graded setting and is central to integration on supermanifolds.

Applications of supermatrices appear in theoretical physics, particularly in supersymmetry and supergravity, as well as in representation theory and supergeometry, where graded linear algebra provides essential tools.