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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.
Eigen is a high-level C++ library of template headers for linear algebra, matrix and vector operations, geometrical transformations, numerical solvers and related algorithms. . Eigen is open-source software licensed under the Mozilla Public License 2.0 since version 3.1
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 ...
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]
A Cartan involution on () is defined by () =, where denotes the transpose matrix of .; The identity map on is an involution. It is the unique Cartan involution of if and only if the Killing form of is negative definite or, equivalently, if and only if is the Lie algebra of a compact semisimple Lie group.
Download QR code; Print/export ... This is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and ...
The algebra E(A) is called the C*-enveloping algebra of the Banach *-algebra A. Of particular importance is the C*-algebra of a locally compact group G. This is defined as the enveloping C*-algebra of the group algebra of G. The C*-algebra of G provides context for general harmonic analysis of G in the case G is non-abelian. In particular, the ...