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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.
An involution is a function f : X → X that, when applied twice, brings one back to the starting point. In mathematics, an involution, involutory function, or self-inverse function [1] is a function f that is its own inverse, f(f(x)) = x. for all x in the domain of f. [2] Equivalently, applying f twice produces the original value.
To construct the inverse P ' of a point P outside a circle Ø: . Draw the segment from O (center of circle Ø) to P.; Let M be the midpoint of OP. (Not shown) Draw the circle c with center M going through P.
Born in the mid '60s for matrix manipulation and still in continuous evolution, it pioneered the most common paradigms of this kind of tools, featuring dynamic typing of the structured data objects, dynamic allocation and garbage collection, operators overloading, dynamic linking of compiled or interpreted additional modules contributed by the ...
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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.
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