Editing Transpose (section)

==== 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>