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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.
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> .
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]
Free Boost C++ template library; binds to optimized BLAS such as the Intel MKL; Includes matrix decompositions, non-linear solvers, and machine learning tooling Eigen: Benoît Jacob C++ 2008 3.4.0 / 08.2021 Free MPL2: Eigen is a C++ template library for linear algebra: matrices, vectors, numerical solvers, and related algorithms. Fastor [5]
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.
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 ...
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