Editing Transpose (section)

=== Definition ===

The transpose of a matrix {{math|'''A'''}}, denoted by {{math|'''A'''<sup>T</sup>}},<ref name="Whitelaw1991">{{cite book|author=T.A. Whitelaw|title=Introduction to Linear Algebra, 2nd edition|url=https://books.google.com/books?id=6M_kDzA7-qIC&q=transpose|date=1 April 1991|publisher=CRC Press|isbn=978-0-7514-0159-2}}</ref> {{math|{{sup|⊤}}'''A'''}}, {{math|'''A'''{{sup|⊤}}}}, <math>A^{\intercal}</math>,<ref>{{Cite web|last=|first=|date=|title=Transpose of a Matrix Product (ProofWiki)|url=https://proofwiki.org/wiki/Transpose_of_Matrix_Product|archive-url=|archive-date=|access-date=4 Feb 2021|website=ProofWiki}}</ref><ref>{{Cite web|date=|title=What is the best symbol for vector/matrix transpose?|url=https://tex.stackexchange.com/questions/30619/what-is-the-best-symbol-for-vector-matrix-transpose|archive-url=|archive-date=|access-date=4 Feb 2021|website=[[Stack Exchange]]}}</ref> {{math|'''A′'''}},<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Transpose|url=https://mathworld.wolfram.com/Transpose.html|access-date=2020-09-08|website=mathworld.wolfram.com|language=en}}</ref> {{math|'''A'''<sup>tr</sup>}}, {{math|<sup>t</sup>'''A'''}} or {{math|'''A'''<sup>t</sup>}}, may be constructed by any one of the following methods:
#[[Reflection (mathematics)|Reflect]] {{math|'''A'''}} over its [[main diagonal]] (which runs from top-left to bottom-right) to obtain {{math|'''A'''<sup>T</sup>}}
#Write the rows of {{math|'''A'''}} as the columns of {{math|'''A'''<sup>T</sup>}}
#Write the columns of {{math|'''A'''}} as the rows of {{math|'''A'''<sup>T</sup>}}

Formally, the {{mvar|i}}-th row, {{mvar|j}}-th column element of {{math|'''A'''<sup>T</sup>}} is the {{mvar|j}}-th row, {{mvar|i}}-th column element of {{math|'''A'''}}:

:<math>\left[\mathbf{A}^\operatorname{T}\right]_{ij} = \left[\mathbf{A}\right]_{ji}.</math>

If {{math|'''A'''}} is an {{math|{{nowrap|''m'' × ''n''}}}} matrix, then {{math|'''A'''<sup>T</sup>}} is an {{math|{{nowrap|''n'' × ''m''}}}} matrix. 

In the case of square matrices, {{math|'''A'''<sup>T</sup>}} may also denote the {{math|T}}th power of the matrix {{math|'''A'''}}. For avoiding a possible confusion, many authors use left upperscripts, that is, they denote the transpose as {{math|<sup>T</sup>'''A'''}}. An advantage of this notation is that no parentheses are needed when exponents are involved: as  {{math|1=({{sup|T}}'''A'''){{sup|''n''}} = {{sup|T}}('''A'''{{sup|''n''}})}}, notation {{math|{{sup|T}}'''A'''{{sup|''n''}}}} is not ambiguous.

In this article, this confusion is avoided by never using the symbol {{math|T}} as a [[variable (mathematics)|variable]] name.

==== Matrix definitions involving transposition ====

A square matrix whose transpose is equal to itself is called a ''[[symmetric matrix]]''; that is, {{math|'''A'''}} is symmetric if
:<math>\mathbf{A}^{\operatorname{T}} = \mathbf{A}.</math>

A square matrix whose transpose is equal to its negative is called a ''[[skew-symmetric matrix]]''; that is, {{math|'''A'''}} is skew-symmetric if
:<math>\mathbf{A}^{\operatorname{T}} = -\mathbf{A}.</math>

A square [[complex number|complex]] matrix whose transpose is equal to the matrix with every entry replaced by its [[complex conjugate]] (denoted here with an overline) is called a ''[[Hermitian matrix]]'' (equivalent to the matrix being equal to its [[conjugate transpose]]); that is, {{math|'''A'''}} is Hermitian if
:<math>\mathbf{A}^{\operatorname{T}} = \overline{\mathbf{A}}.</math>

A square [[complex number|complex]] matrix whose transpose is equal to the negation of its complex conjugate is called a ''[[skew-Hermitian matrix]]''; that is, {{math|'''A'''}} is skew-Hermitian if
:<math>\mathbf{A}^{\operatorname{T}} = -\overline{\mathbf{A}}.</math>

A square matrix whose transpose is equal to its [[Inverse matrix|inverse]] is called an ''[[orthogonal matrix]]''; that is, {{math|'''A'''}} is orthogonal if
:<math>\mathbf{A}^{\operatorname{T}} = \mathbf{A}^{-1}.</math>

A square complex matrix whose transpose is equal to its conjugate inverse is called a ''[[unitary matrix]]''; that is, {{math|'''A'''}} is unitary if
:<math>\mathbf{A}^{\operatorname{T}} = \overline{\mathbf{A}^{-1}}.</math>