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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 S is a commutative semigroup then the identity map of S is an involution.; If S is a group then the inversion map * : S → S defined by x* = x −1 is an involution. Furthermore, on an abelian group both this map and the one from the previous example are involutions satisfying the axioms of semigroup with involution.
Expression templates have been found especially useful by the authors of libraries for linear algebra, that is, for dealing with vectors and matrices of numbers. Among libraries employing expression template are Dlib , Armadillo , Blaze , [ 5 ] Blitz++ , [ 6 ] Boost uBLAS, [ 7 ] Eigen , [ 8 ] POOMA, [ 9 ] Stan Math Library , [ 10 ] and xtensor ...
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 ...
To change this template's initial visibility, the |state= parameter may be used: {{Linear algebra | state = collapsed}} will show the template collapsed, i.e. hidden apart from its title bar. {{Linear algebra | state = expanded}} will show the template expanded, i.e. fully visible.
The Matrix Template Library (MTL) is a linear algebra library for C++ programs. The MTL uses template programming , which considerably reduces the code length. All matrices and vectors are available in all classical numerical formats: float , double , complex<float> or complex<double> .
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The algebra structure over the complex numbers is obtained by considering the pointwise operations of addition and multiplication. The involution is pointwise complex conjugation. The norm is the uniform norm on functions. The importance of X being locally compact and Hausdorff is that this turns X into a completely regular space.
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