Editing Involution (mathematics) (section)

=== Linear algebra ===
{{details|Involutory matrix}}
In linear algebra, an involution is a linear operator {{math|''T''}} on a vector space, such that {{math|1={{itco|''T''}}{{sup|2}} = ''I''}}. Except for in characteristic 2, such operators are diagonalizable for a given basis with just {{math|1}}s and {{math|&minus;1}}s on the diagonal of the corresponding matrix. If the operator is orthogonal (an '''orthogonal involution'''), it is orthonormally diagonalizable.

For example, suppose that a basis for a vector space {{math|''V''}} is chosen, and that {{math|''e''<sub>1</sub>}} and {{math|''e''<sub>2</sub>}} are basis elements. There exists a linear transformation {{math|''f''}} that sends {{math|''e''<sub>1</sub>}} to {{math|''e''<sub>2</sub>}}, and sends {{math|''e''<sub>2</sub>}} to {{math|''e''<sub>1</sub>}}, and that is the identity on all other basis vectors. It can be checked that {{math|1=''f''(''f''(''x'')) = ''x''}} for all {{math|''x''}} in {{math|''V''}}. That is, {{math|''f''}} is an involution of {{math|''V''}}.

For a specific basis, any linear operator can be represented by a [[matrix (mathematics)|matrix]] {{math|''T''}}. Every matrix has a [[transpose]], obtained by swapping rows for columns. This transposition is an involution on the set of matrices.  Since elementwise [[complex conjugation]] is an independent involution, the [[conjugate transpose]] or [[Hermitian adjoint]] is also an involution.

The definition of involution extends readily to [[module (mathematics)|modules]]. Given a module {{math|''M''}} over a [[ring (mathematics)|ring]] {{math|''R''}}, an {{math|''R''}} [[endomorphism]] {{math|''f''}} of {{math|''M''}} is called an involution if {{math|{{itco|''f''}}<sup>2</sup>}} is the identity homomorphism on {{math|''M''}}.

[[Idempotent element (ring theory)#Relation with involutions|Involutions are related to idempotent]]s; if {{math|2}} is invertible then they [[bijection|correspond]] in a one-to-one manner.

In [[functional analysis]], [[Banach *-algebra]]s and [[C*-algebra]]s are special types of [[Banach algebra]]s with involutions.