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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.
An involution is non-defective, and each eigenvalue equals , so an involution diagonalizes to a signature matrix. A normal involution is Hermitian (complex) or symmetric (real) and also unitary (complex) or orthogonal (real). The determinant of an involutory matrix over any field is ±1. [4]
An example from linear algebra is the multiplicative monoid of real square matrices of order n (called the full linear monoid). The map which sends a matrix to its transpose is an involution because the transpose is well defined for any matrix and obeys the law (AB) T = B T A T, which has the same form of interaction with multiplication as ...
In this example, a self-adjoint morphism is a symmetric relation. The category Cob of cobordisms is a dagger compact category , in particular it possesses a dagger structure. The category Hilb of Hilbert spaces also possesses a dagger structure: Given a bounded linear map f : A → B {\displaystyle f:A\rightarrow B} , the map f † : B → A ...
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).
The only linear function satisfying this condition is a constant (conventionally 1) times B, so Berezin defined [7] ∫ d θ ( A + B θ ) ≡ B . {\displaystyle \int d\theta (A+B\theta )\equiv B.} This results in the following rules for the integration of a Grassmann quantity:
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