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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.
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.
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]
Functional analysis applies the methods of linear algebra alongside those of mathematical analysis to study various function spaces; the central objects of study in functional analysis are L p spaces, which are Banach spaces, and especially the L 2 space of square-integrable functions, which is the only Hilbert space among them. Functional ...
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:
For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number ...
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.
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