Editing Transpose (section)

=== Adjoint ===
{{distinguish|Hermitian adjoint}}

If the vector spaces {{mvar|X}} and {{mvar|Y}} have respectively [[nondegenerate form|nondegenerate]] [[bilinear form]]s {{math|''B''<sub>''X''</sub>}} and {{math|''B''<sub>''Y''</sub>}}, a concept known as the '''adjoint''', which is closely related to the transpose, may be defined:

If {{nowrap|{{math|''u'' : ''X'' → ''Y''}}}} is a [[linear map]] between [[vector space]]s {{mvar|X}} and {{mvar|Y}}, we define {{mvar|g}} as the '''adjoint''' of {{mvar|u}} if {{nowrap|{{math|''g'' : ''Y'' → ''X''}}}} satisfies
:<math>B_X\big(x, g(y)\big) = B_Y\big(u(x), y\big)</math> for all {{math|''x'' &isin; ''X''}} and {{math|''y'' &isin; ''Y''}}.

These bilinear forms define an [[isomorphism]] between {{mvar|X}} and {{math|''X''<sup>#</sup>}}, and between {{mvar|''Y''}} and {{math|''Y''<sup>#</sup>}}, resulting in an isomorphism between the transpose and adjoint of {{mvar|u}}. 
The matrix of the adjoint of a map is the transposed matrix only if the [[basis (linear algebra)|bases]] are [[Orthonormality|orthonormal]] with respect to their bilinear forms. 
In this context, many authors however, use the term transpose to refer to the adjoint as defined here.

The adjoint allows us to consider whether {{nowrap|{{math|''g'' : ''Y'' → ''X''}}}} is equal to {{nowrap|{{math|''u''<sup> −1</sup> : ''Y'' → ''X''}}}}. 
In particular, this allows the [[orthogonal group]] over a vector space {{mvar|X}} with a quadratic form to be defined without reference to matrices (nor the components thereof) as the set of all linear maps {{nowrap|{{math|''X'' → ''X''}}}} for which the adjoint equals the inverse.

Over a complex vector space, one often works with [[sesquilinear form]]s (conjugate-linear in one argument) instead of bilinear forms. 
The [[Hermitian adjoint]] of a map between such spaces is defined similarly, and the matrix of the Hermitian adjoint is given by the conjugate transpose matrix if the bases are orthonormal.