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An involution is a function f : X → X that, when applied twice, brings one back to the starting point. In mathematics, an involution, involutory function, or self-inverse function [1] is a function f that is its own inverse, f(f(x)) = x. for all x in the domain of f. [2] Equivalently, applying f twice produces the original value.
If A represents a linear involution, then x→A(x−b)+b is an affine involution. One can check that any affine involution in fact has this form. Geometrically this means that any affine involution can be obtained by taking oblique reflections against any number from 0 through n hyperplanes going through a point b.
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients [1]: ch. 17 [2]: ch. 10 (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence.
A linear program may be specified by a system of real variables (the coordinates for a point in Euclidean space ), a system of linear constraints (specifying that the point lie in a halfspace; the intersection of these halfspaces is a convex polytope, the feasible region of the program), and a linear function (what to optimize). Every linear ...
In terms of linear algebra, assuming the origin is fixed, involutions are exactly the diagonalizable maps with all eigenvalues either 1 or −1. Reflection in a hyperplane has a single −1 eigenvalue (and multiplicity n − 1 {\displaystyle n-1} on the 1 eigenvalue), while point reflection has only the −1 eigenvalue (with multiplicity n ).
In a dagger category , a morphism is called . unitary if † =,; self-adjoint if † =.; The latter is only possible for an endomorphism:.The terms unitary and self-adjoint in the previous definition are taken from the category of Hilbert spaces, where the morphisms satisfying those properties are then unitary and self-adjoint in the usual sense.
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