Supermatrix

In mathematics and theoretical physics, a supermatrix is a Z₂-graded analog of an ordinary matrix. Specifically, a supermatrix is a 2×2 block matrix with entries in a superalgebra (or superring). The most important examples are those with entries in a commutative superalgebra (such as a Grassmann algebra) or an ordinary field (thought of as a purely even commutative superalgebra).

Supermatrices arise in the study of super linear algebra where they appear as the coordinate representations of a linear transformations between finite-dimensional super vector spaces or free supermodules. They have important applications in the field of supersymmetry.

Let R be a fixed superalgebra (assumed to be unital and associative). Often one requires R be supercommutative as well (for essentially the same reasons as in the ungraded case).

Let p, q, r, and s be nonnegative integers. A supermatrix of dimension (r|s)×(p|q) is a matrix with entries in R that is partitioned into a 2×2 block structure

with r+s total rows and p+q total columns (so that the submatrix X₀₀ has dimensions r×p and X₁₁ has dimensions s×q). An ordinary (ungraded) matrix can be thought of as a supermatrix for which q and s are both zero.

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