Given objects and in an additive category, we can represent morphisms as -by- matrices :\begin{pmatrix} f_{11} & f_{12} & \cdots & f_{1n} \\ f_{21} & f_{22} & \cdots & f_{2n} \\ \vdots & \vdots & \cdots & \vdots \\ f_{m1} & f_{m2} & \cdots & f_{mn} \end{pmatrix} where f_{kl} := p_k \circ f \circ i_l\colon A_l \to B_k. Using that , it follows that addition and composition of matrices obey the usual rules for
matrix addition and
multiplication. n
to be the n
-fold biproduct A
⊕ ⋯ ⊕ A
and Bm
similarly, then the morphisms from An
to Bm
can be described as m
-by-n
matrices whose entries are morphisms from A
to B''. For a concrete example, consider the category of
real vector spaces, so that
A and
B are individual vector spaces. (There is no need for
A and
B to have
finite dimension (mathematics)|s, although of course the numbers
m and
n must be finite.) Then an element of
An may be represented as an
n-by-
column vector whose entries are elements of
A: \begin{pmatrix} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{pmatrix} and a morphism from
An to
Bm is an
m-by-
n matrix whose entries are morphisms from
A to
B: \begin{pmatrix} f_{1,1} & f_{1,2} & \cdots & f_{1,n} \\ f_{2,1} & f_{2,2} & \cdots & f_{2,n} \\ \vdots & \vdots & \cdots & \vdots \\ f_{m,1} & f_{m,2} & \cdots & f_{m,n} \end{pmatrix} Then this morphism matrix acts on the column vector by the usual rules of matrix multiplication to give an element of
Bm, represented by an
m-by-1 column vector with entries from
B: \begin{pmatrix} f_{1,1} & f_{1,2} & \cdots & f_{1,n} \\ f_{2,1} & f_{2,2} & \cdots & f_{2,n} \\ \vdots & \vdots & \cdots & \vdots \\ f_{m,1} & f_{m,2} & \cdots & f_{m,n} \end{pmatrix} \begin{pmatrix} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{pmatrix} = \begin{pmatrix} f_{1,1}(a_{1}) + f_{1,2}(a_{2}) + \cdots + f_{1,n}(a_{n}) \\ f_{2,1}(a_{1}) + f_{2,2}(a_{2}) + \cdots + f_{2,n}(a_{n}) \\ \cdots \\ f_{m,1}(a_{1}) + f_{m,2}(a_{2}) + \cdots + f_{m,n}(a_{n}) \end{pmatrix} Even in the setting of an abstract additive category, where it makes no sense to speak of elements of the objects
An and
Bm, the matrix representation of the morphism is still useful, because
matrix multiplication correctly reproduces composition of morphisms. --> Thus additive categories can be seen as the most general context in which the algebra of matrices makes sense. Recall that the morphisms from a single object to itself form the
endomorphism ring . If we denote the -fold product of with itself by , then morphisms from to are
m-by-
n matrices with entries from the ring . Conversely, given any
ring , we can form a category by taking objects
An indexed by the set of
natural numbers (including
0) and letting the hom-set of morphisms from to be the
set of -by- matrices over , and where composition is given by matrix multiplication. Then is an additive category, and equals the -fold power . This construction should be compared with the result that a ring is a preadditive category with just one object, shown
here. If we interpret the object as the left
module , then this
matrix category becomes a
subcategory of the category of left modules over . This may be confusing in the special case where or is zero, because we usually don't think of
matrices with 0 rows or 0 columns. This concept makes sense, however: such matrices have no entries and so are completely determined by their size. While these matrices are rather degenerate, they do need to be included to get an additive category, since an additive category must have a zero object. Thinking about such matrices can be useful in one way, though: they highlight the fact that given any objects and in an additive category, there is exactly one morphism from to 0 (just as there is exactly one 0-by-1 matrix with entries in ) and exactly one morphism from 0 to (just as there is exactly one 1-by-0 matrix with entries in ) – this is just what it means to say that 0 is a
zero object. Furthermore, the zero morphism from to is the composition of these morphisms, as can be calculated by multiplying the degenerate matrices. == Additive functors ==