In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse,
for all in the domain of . Equivalently, applying twice produces the original value.
General properties
Any involution is a bijection.
The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (), reciprocation (), and complex conjugation () in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher.
The composition of two involutions f and g is an involution if and only if they commute: .
Involutions on finite sets
The number of involutions, including the identity involution, on a set with elements is given by a recurrence relation found by Heinrich August Rothe in 1800: and for The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 ; these numbers are called the telephone numbers, and they also count the number of Young tableaux with a given number of cells. The number can also be expressed by non-recursive formulas, such as the sum
The number of fixed points of an involution on a finite set and its number of elements have the same parity. Thus the number of fixed points of all the involutions on a given finite set have the same parity. In particular, every involution on an odd number of elements has at least one fixed point. This can be used to prove Fermat's two squares theorem.
Involution throughout the fields of mathematics
Real-valued functions
Some basic examples of involutions include the functions
the composition
and more generally the function
is an involution for constants and that satisfy
Another one is
The graph of an involution (on the real numbers) is symmetric across the line . This is due to the fact that the inverse of any general function will be its reflection over the line . This can be seen by "swapping" with . If, in particular, the function is an involution, then its graph is its own reflection.
Other elementary involutions are useful in solving functional equations.
Euclidean geometry
A simple example of an involution of the three-dimensional Euclidean space is reflection through a plane. Performing a reflection twice brings a point back to its original coordinates.
Another involution is reflection through the origin; not a reflection in the above sense, and so, a distinct example.
These transformations are examples of affine involutions.
Projective geometry
An involution is a projectivity of period 2, that is, a projectivity that interchanges pairs of points. Any projectivity that interchanges two points is an involution. The three pairs of opposite sides of a complete quadrangle meet any line (not through a vertex) in three pairs of an involution. This theorem has been called Desargues's Involution Theorem. Its origins can be seen in Lemma IV of the lemmas to the Porisms of Euclid in Volume VII of the Collection of Pappus of Alexandria. If an involution has one fixed point, it has another, and consists of the correspondence between harmonic conjugates with respect to these two points. In this instance the involution is termed "hyperbolic", while if there are no fixed points it is "elliptic". In the context of projectivities, fixed points are called double points.
Another type of involution occurring in projective geometry is a polarity which is a correlation of period 2.
Linear algebra
In linear algebra, an involution is a linear operator T on a vector space, such that . Except for in characteristic 2, such operators are diagonalizable for a given basis with just 1s and −1s 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 V is chosen, and that e1 and e2 are basis elements. There exists a linear transformation f which sends e1 to e2, and sends e2 to e1, and which is the identity on all other basis vectors. It can be checked that for all x in V. That is, f is an involution of V.
For a specific basis, any linear operator can be represented by a matrix 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.