Editing Division (mathematics) (section)

=== Of matrices ===
One can define a division operation for matrices. The usual way to do this is to define {{nowrap|1=''A'' / ''B'' = ''AB''<sup>−1</sup>}}, where {{nowrap|''B''<sup>−1</sup>}} denotes the [[inverse matrix|inverse]] of ''B'', but it is far more common to write out {{nowrap|''AB''<sup>−1</sup>}} explicitly to avoid confusion. An [[elementwise division]] can also be defined in terms of the [[Hadamard product (matrices)|Hadamard product]].

==== Left and right division ====
Because [[matrix multiplication]] is not [[commutative]], one can also define a [[left division]] or so-called ''backslash-division'' as {{math|1=''A'' \ ''B'' = ''A''<sup>−1</sup>''B''}}. For this to be well defined, {{math|''B''<sup>−1</sup>}} need not exist, however {{math|''A''<sup>−1</sup>}} does need to exist. To avoid confusion, division as defined by {{math|1=''A'' / ''B'' = ''AB''<sup>−1</sup>}} is sometimes called ''right division'' or ''slash-division'' in this context.

With left and right division defined this way, {{math|''A'' / (''BC'')}} is in general not the same as {{math|(''A'' / ''B'') / ''C''}}, nor is {{math|(''AB'') \ ''C''}} the same as {{math|''A'' \ (''B'' \ ''C'')}}. However, it holds that {{math|1=''A'' / (''BC'') = (''A'' / ''C'') / ''B''}} and {{math|1=(''AB'') \ ''C'' = ''B'' \ (''A'' \ ''C'')}}.

==== Pseudoinverse ====
To avoid problems when {{math|''A''<sup>−1</sup>}} and/or {{math|''B''<sup>−1</sup>}} do not exist, division can also be defined as multiplication by the [[Generalized inverse|pseudoinverse]]. That is, {{math|1=''A'' / ''B'' = ''AB''<sup>+</sup>}} and {{math|1=''A'' \ ''B'' = ''A''<sup>+</sup>''B''}}, where {{math|''A''<sup>+</sup>}} and {{math|''B''<sup>+</sup>}} denote the pseudoinverses of {{mvar|A}} and {{mvar|B}}.