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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]
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.
Any involution is a bijection.. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x ↦ −x), reciprocation (x ↦ 1/x), and complex conjugation (z ↦ z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the ...
Download QR code; Print/export ... (i.e. an involution). ... a map is anticonformal if at every point the Jacobian is a scalar times an orthogonal matrix with ...
As the involution is antilinear, it cannot be the identity map on . Of course, φ {\textstyle \varphi } is a R {\textstyle \mathbb {R} } -linear transformation of V , {\textstyle V,} if one notes that every complex space V {\displaystyle V} has a real form obtained by taking the same vectors as in the original space and restricting the scalars ...
In this construction, A is an algebra with involution, meaning: A is an abelian group under + A has a product that is left and right distributive over + A has an involution *, with (x*)* = x, (x + y)* = x* + y*, (xy)* = y*x*. The algebra B = A ⊕ A produced by the Cayley–Dickson construction is also an algebra with involution.
Download QR code; Print/export ... where T is an infinite-dimensional operator with matrix elements T nk. The transform is an involution, ...
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