Editing Transpose (section)

=== Products ===
If {{math|'''A'''}} is an {{math|{{nowrap|''m'' × ''n''}}}} matrix and {{math|'''A'''<sup>T</sup>}} is its transpose, then the result of [[matrix multiplication]] with these two matrices gives two square matrices: {{math|'''A A'''<sup>T</sup>}} is {{math|{{nowrap|''m'' × ''m''}}}} and {{math|'''A'''<sup>T</sup> '''A'''}} is {{math|{{nowrap|''n'' × ''n''}}}}. Furthermore, these products are [[symmetric matrices]]. Indeed, the matrix product {{math|'''A A'''<sup>T</sup>}} has entries that are the [[inner product]] of a row of {{math|'''A'''}} with a column of {{math|'''A'''<sup>T</sup>}}. But the columns of {{math|'''A'''<sup>T</sup>}} are the rows of {{math|'''A'''}}, so the entry corresponds to the inner product of two rows of {{math|'''A'''}}. If {{mvar|p<sub>i j</sub>}} is the entry of the product, it is obtained from rows {{mvar|i}} and {{mvar|j}} in {{math|'''A'''}}. The entry {{mvar|p<sub>j i</sub>}} is also obtained from these rows, thus {{math|''p''<sub>i j</sub> {{=}} ''p''<sub>j i</sub>}}, and the product matrix ({{mvar|p<sub>i j</sub>}}) is symmetric. Similarly, the product {{math|'''A'''<sup>T</sup> '''A'''}} is a symmetric matrix.

A quick proof of the symmetry of {{math|'''A A'''<sup>T</sup>}} results from the fact that it is its own transpose:
:<math>\left(\mathbf{A} \mathbf{A}^\operatorname{T}\right)^\operatorname{T} = \left(\mathbf{A}^\operatorname{T}\right)^\operatorname{T} \mathbf{A}^\operatorname{T}= \mathbf{A} \mathbf{A}^\operatorname{T} .</math><ref>[[Gilbert Strang]] (2006) ''Linear Algebra and its Applications'' 4th edition, page 51, Thomson [[Brooks/Cole]] {{ISBN|0-03-010567-6}}</ref>