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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.
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
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]
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 terms of linear algebra, assuming the origin is fixed, involutions are exactly the diagonalizable maps with all eigenvalues either 1 or −1. Reflection in a hyperplane has a single −1 eigenvalue (and multiplicity n − 1 {\displaystyle n-1} on the 1 eigenvalue), while point reflection has only the −1 eigenvalue (with multiplicity n ).
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.
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