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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]
Every matrix has a transpose, obtained by swapping rows for columns. This transposition is an involution on the set of matrices. Since elementwise complex conjugation is an independent involution, the conjugate transpose or Hermitian adjoint is also an involution. The definition of involution extends readily to modules.
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.
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.
In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication.The unitary group is a subgroup of the general linear group GL(n, C), and it has as a subgroup the special unitary group, consisting of those unitary matrices with determinant 1.
In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, considered as unary operator, exhibits certain fundamental properties of the operation of taking the inverse in a group:
It is also closed under involution; hence it is a C*-algebra. Concrete C*-algebras of compact operators admit a characterization similar to Wedderburn's theorem for finite dimensional C*-algebras: Theorem. If A is a C*-subalgebra of K(H), then there exists Hilbert spaces {H i} i∈I such that
In formal mathematical presentations of T-symmetry, three different kinds of notation for T need to be carefully distinguished: the T that is an involution, capturing the actual reversal of the time coordinate, the T that is an ordinary finite dimensional matrix, acting on spinors and vectors, and the T that is an operator on an infinite ...