As Linear Transformations
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). There is an important difference in the graded case, however. A homomorphism from one super vector space to another is, by definition, one that preserves the grading (i.e. maps even elements to even elements and odd elements to odd elements). The coordinate representation of such a transformation is always an even supermatrix. Odd supermatrices correspond to linear transformations that reverse the grading. General supermatrices represent an arbitrary ungraded linear transformation. Such transformations are still important in the graded case, although less so than the graded (even) transformations.
A supermodule M over a superalgebra R is free if it has a free homogeneous basis. If such a basis consists of p even elements and q odd elements, then M is said to have rank p|q. If R is supercommutative, the rank is independent of the choice of basis, just as in the ungraded case.
Let Rp|q be the space of column supervectors—supermatrices of dimension (p|q)×(1|0). This is naturally a right R-supermodule, called the right coordinate space. A supermatrix T of dimension (r|s)×(p|q) can then be thought of as a right R-linear map
where the action of T on Rp|q is just supermatrix multiplication (this action is not generally left R-linear which is why we think of Rp|q as a right supermodule).
Let M be free right R-supermodule of rank p|q and let N be a free right R-supermodule of rank r|s. Let (ei) be a free basis for M and let (fk) be a free basis for N. Such a choice of bases is equivalent to a choice of isomorphisms from M to Rp|q and from N to Rr|s. 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:
Read more about this topic: Supermatrix