Involutive

Involutive describes a transformation or operation in mathematics that, when applied twice, yields the original object. More formally, a map f: X → X is involutive if f(f(x)) = x for every x in X. Equivalently, f is its own inverse, so f = f^{-1} and f^2 is the identity on X. If f is linear, this means f^2 = I.

Common examples include complex conjugation on the complex numbers, f(z) = z̄, since applying it twice returns z. Geometric reflections are involutive: reflecting across a line and then reflecting again returns the original point. The negation map x ↦ -x is involutive on any vector space. In the symmetric group, a transposition is an involution, and a product of disjoint transpositions is also an involution.

In linear algebra and representation theory, an operator T with T^2 = I is an involution; its eigenvalues are ±1 and the space splits into the direct sum of its (+1)- and (-1)-eigenspaces. In group theory, an element g with g^2 = e is called an involution. In the context of algebras, an involution (or star operation) is an involutive anti-automorphism, denoted *, satisfying (ab)^* = b^* a^* and (a^*)^* = a; many algebraic structures, including C*-algebras, use such an involution.

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