Editing Additive category (section)

== Matrix representation of morphisms ==

Given objects {{math|''A''<sub>1</sub>, ..., ''A<sub>n</sub>''}} and {{math|''B''<sub>1</sub>, ..., ''B<sub>m</sub>''}} in an additive category, we can represent morphisms {{math|''f'': ''A''<sub>1</sub> ⊕ ⋅⋅⋅ ⊕ ''A<sub>n</sub>'' → ''B''<sub>1</sub> ⊕ ⋅⋅⋅ ⊕ ''B<sub>m</sub>''}} as {{mvar|m}}-by-{{mvar|n}} matrices
:<math>\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}
</math> where <math>f_{kl} := p_k \circ f \circ i_l\colon A_l \to B_k.</math>

Using that {{math|1=∑<sub>''k''</sub> ''i''<sub>''k''</sub> ∘ ''p''<sub>''k''</sub> = 1}}, it follows that addition and composition of matrices obey the usual rules for [[matrix addition]] and [[matrix multiplication|multiplication]].

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use [[matrix (mathematics)|matrices]] to study the biproducts of ''A'' and ''B'' with themselves.
Specifically, if we define the ''biproduct power'' ''A<sup>n</sup>'' to be the ''n''-fold biproduct ''A'' ⊕ ⋯ ⊕ ''A'' and ''B<sup>m</sup>'' similarly, then the morphisms from ''A<sup>n</sup>'' to ''B<sup>m</sup>'' 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 number|real]] [[vector space]]s, so that ''A'' and ''B'' are individual vector spaces.
(There is no need for ''A'' and ''B'' to have [[finite set|finite]] [[dimension (mathematics)|]]s, although of course the numbers ''m'' and ''n'' must be finite.)
Then an element of ''A''<sup>''n''</sup> may be represented as an ''n''-by-{{num|1}} [[column vector]] whose entries are elements of ''A'':

<math>\begin{pmatrix} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{pmatrix}</math>

and a morphism from ''A''<sup>''n''</sup> to ''B''<sup>''m''</sup> is an ''m''-by-''n'' matrix whose entries are morphisms from ''A'' to ''B'':

<math>\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}</math>

Then this morphism matrix acts on the column vector by the usual rules of matrix multiplication to give an element of ''B''<sup>''m''</sup>, represented by an ''m''-by-1 column vector with entries from ''B'':

<math>\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}</math>

Even in the setting of an abstract additive category, where it makes no sense to speak of elements of the objects ''A''<sup>''n''</sup> and ''B''<sup>''m''</sup>, the matrix representation of the morphism is still useful, because [[matrix multiplication]] correctly reproduces composition of morphisms.
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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&nbsp;{{mvar|A}} to itself form the [[endomorphism ring]] {{math|End&thinsp;''A''}}.
If we denote the {{mvar|n}}-fold product of&nbsp;{{mvar|A}} with itself by {{math|''A''<sup>''n''</sup>}}, then morphisms from {{math|''A<sup>n</sup>''}} to {{math|''A<sup>m</sup>''}} are ''m''-by-''n'' matrices with entries from the ring {{math|End&thinsp;''A''}}.

Conversely, given any [[ring (mathematics)|ring]] {{mvar|R}}, we can form a category&nbsp;{{math|'''Mat'''(''R'')}} by taking objects ''A<sub>n</sub>'' indexed by the set of [[natural number]]s (including [[0]]) and letting the hom-set of morphisms from {{math|''A<sub>n</sub>''}} to {{math|''A<sub>m</sub>''}} be the [[set (mathematics)|set]] of {{mvar|m}}-by-{{mvar|n}} matrices over&nbsp;{{mvar|R}}, and where composition is given by matrix multiplication.<ref>H.D. Macedo, J.N. Oliveira, [https://hal.inria.fr/hal-00919866 Typing linear algebra: A biproduct-oriented approach], Science of Computer Programming, Volume 78, Issue 11, 1 November 2013, Pages 2160-2191, {{issn|0167-6423}}, {{doi|10.1016/j.scico.2012.07.012}}.</ref> Then {{math|'''Mat'''(''R'')}} is an additive category, and {{math|''A''<sub>''n''</sub>}} equals the {{mvar|n}}-fold power {{math|(''A''<sub>1</sub>)<sup>''n''</sup>}}.

This construction should be compared with the result that a ring is a preadditive category with just one object, shown [[Preadditive category#Special cases|here]].

If we interpret the object {{math|''A''<sub>''n''</sub>}} as the left [[module (mathematics)|module]]&nbsp;{{math|''R''<sup>''n''</sup>}}, then this ''matrix category'' becomes a [[subcategory]] of the category of left modules over&nbsp;{{mvar|R}}.

This may be confusing in the special case where {{mvar|m}} or {{mvar|n}} is zero, because we usually don't think of [[empty matrix|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 {{mvar|A}} and {{mvar|B}} in an additive category, there is exactly one morphism from {{mvar|A}} to 0 (just as there is exactly one 0-by-1 matrix with entries in {{math|End&thinsp;''A''}}) and exactly one morphism from 0 to {{mvar|B}} (just as there is exactly one 1-by-0 matrix with entries in {{math|End&thinsp;''B''}})&nbsp;– this is just what it means to say that 0 is a [[zero object (algebra)|zero object]]. Furthermore, the zero morphism from {{mvar|A}} to {{mvar|B}} is the composition of these morphisms, as can be calculated by multiplying the degenerate matrices.