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).
These operations are exactly the same as the ordinary ones with the restriction that they are defined only when the blocks have compatible dimensions.
This set forms a supermodule over R under supermatrix addition and scalar multiplication.
In particular, if R is a superalgebra over a field K then Mr|s×p|q(R) forms a super vector space over K. Let Mp|q(R) denote the set of all square supermatices over R with dimension (p|q)×(p|q).
The multiplication can be performed at the block level in the obvious manner: Note that the blocks of the product supermatrix Z = XY are given by If X and Y are homogeneous with parities |X| and |Y| then XY is homogeneous with parity |X| + |Y|.
Scalar multiplication for supermatrices is different than the ungraded case due to the presence of odd elements in R. Let X be a supermatrix.
Furthermore, if R is supercommutative then one has Ordinary matrices can be thought of as the coordinate representations of linear maps between vector spaces (or free modules).
Likewise, supermatrices can be thought of as the coordinate representations of linear maps between super vector spaces (or free supermodules).
If R is supercommutative, the rank is independent of the choice of basis, just as in the ungraded case.
Any (ungraded) linear map can be written as a (r|s)×(p|q) supermatrix relative to the chosen bases.
The components of the associated supermatrix are determined by the formula The block decomposition of a supermatrix T corresponds to the decomposition of M and N into even and odd submodules: Many operations on ordinary matrices can be generalized to supermatrices, although the generalizations are not always obvious or straightforward.
Unlike the ordinary transpose, the supertranspose is not generally an involution, but rather has order 4.
It is defined on homogeneous supermatrices by the formula where tr denotes the ordinary trace.