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In linear algebra, an involution is a linear operator T on a vector space, such that T 2 = I. Except for in characteristic 2, such operators are diagonalizable for a given basis with just 1 s and −1 s on the diagonal of the corresponding matrix. If the operator is orthogonal (an orthogonal involution), it is orthonormally diagonalizable.
In a dagger category , a morphism is called . unitary if † =,; self-adjoint if † =.; The latter is only possible for an endomorphism:.The terms unitary and self-adjoint in the previous definition are taken from the category of Hilbert spaces, where the morphisms satisfying those properties are then unitary and self-adjoint in the usual sense.
A *-algebra A is a *-ring, [b] with involution * that is an associative algebra over a commutative *-ring R with involution ′, such that (r x)* = r ′ x* ∀r ∈ R, x ∈ A. [3] The base *-ring R is often the complex numbers (with ′ acting as complex conjugation). It follows from the axioms that * on A is conjugate-linear in R, meaning
In mathematics, specifically in functional analysis, a C ∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:
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]
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; ... The operation of taking the transpose is an involution ...
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.
Involution (mathematics), a function that is its own inverse; Involution algebra, a *-algebra: a type of algebraic structure; Involute, a construction in the differential geometry of curves; Exponentiation (archaic use of the term)
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