Selfinverse

Selfinverse is a term used in mathematics to describe an object that is its own inverse. In particular, a function f is selfinverse if applying it twice returns the original input: f(f(x)) = x for all x in its domain. Equivalently, f equals its inverse function f^{-1}. Such functions are commonly called involutions, and the phrase self-inverse is often used interchangeably.

Common examples include the identity function f(x) = x, which trivially satisfies f(f(x)) = x; the negation f(x) = -x on the real numbers; and the reciprocal f(x) = 1/x on the domain R \ {0}, since f(f(x)) = x in that domain. In group theory, an element g is self-inverse if g^{-1} = g, which means g^2 is the identity element.

In the language of permutations, a permutation is self-inverse iff it is composed of disjoint transpositions and fixed points. In linear algebra, a matrix A is self-inverse if A^2 = I. Examples include reflection matrices (such as diag(1, -1) in R^2) and the scalar matrix -I. More generally, any linear transformation that represents a reflection across a subspace is self-inverse.

Properties and considerations vary by context, but a key feature is that applying a self-inverse map twice recovers the starting object. The study of self-inverse operations intersects with symmetry, functional equations, and various areas of algebra and geometry.

See also: involution, inverse function, permutation, symmetry.