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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]
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 ...
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.
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 ...
for the transformation, where T is an infinite-dimensional operator with matrix elements T nk. The transform is an involution, that is, = or, using index notation, = = where is the Kronecker delta. The original series can be regained by
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
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.
Because the bit-reversal permutation is an involution, it may be performed easily in place (without copying the data into another array) by swapping pairs of elements. In the random-access machine commonly used in algorithm analysis, a simple algorithm that scans the indexes in input order and swaps whenever the scan encounters an index whose ...