Hypercomplex number
Hyper Complex numbers are generalizations of complex numbers. In this article, hypercomplex numbers are considered as algebraic structure. Sometimes, the quaternions are referred to as the hypercomplex numbers.
Definition
A hyper- complex number is an element of an algebra hyper complex numbers. An algebra over the real numbers is called algebra hypercomplex numbers or hypercomplex system of rank, if
- It as a vector space has finite dimension and if
- It has a unit element, that is, if one exists, such that the equation holds for all.
Some authors require in addition that the algebra with respect to multiplication is associative. In particular, the real numbers are even an algebra hyper complex numbers.
Properties
- For the addition of the commutative and associative law apply.
- The addition is invertible.
- The left-hand and right-hand distributive law.
- The multiplication in a hyper-complex algebra A is bilinear over the real numbers, ie it is
- For the multiplication of the commutative hypercomplex numbers need not apply.
- The multiplication need not be zero divisors.
Conjugation
Hyper Complex numbers can be represented as follows as the sum of:
- .
The sizes for k> 0 are called imaginary units. The conjugated to number produced by all imaginary units are replaced by their negative (). To the conjugated complex number is represented by or. Their sum representation is
- .
Conjugation is an involution of the hypercomplex numbers, that is,
- .
Examples
Complex Numbers
The complex numbers are a hypercomplex number system by
- With
Is defined.
The binary numbers are defined by
- With.
Dual numbers
The dual numbers are defined by
- With.
Note that they have nothing to do with binary numbers.
Quaternions
The quaternions (symbol often after their discoverer, WR Hamilton ) form a four-dimensional algebra with division and associative (but not commutative ) multiplication. So it is with the quaternions a skew field.
Biquaternionen
Are defined as the Biquaternionen quaternions with complex coefficients, i.e., they form a four-dimensional vector space as well as the quaternion form a four-dimensional vector space.
Octonions
The octonions (symbol, also called octaves) are eight-dimensional hypercomplex numbers with division and multiplication alternative.
Sedenionen
The Sedenionen (symbol ) are sechzehndimensionale hypercomplex numbers. Your multiplication is not commutative, associative, nor alternatively. Also they have no division; Instead, they have zero divisors.
Square matrices
Let n be a natural number. The is then an algebra with the identity matrix as one element - ie a hypercomplex algebra. More particularly, it is an associative hypercomplex algebra and thus, a ring, and as such also unitary. The reellzahligen multiples of the unit matrix form a subalgebra isomorphic to.
In case there are subalgebras which are isomorphic to the above 3 -dimensional algebras; they are distinguished by the fact that the main diagonal elements are always the same (which corresponds to the real part ) and apply the elements of the secondary diagonals rules that define the algebra shown:
- An off-diagonal element is 0 → The algebra is isomorphic to the dual numbers
- Both off-diagonal elements are the same → The algebra is isomorphic to the binary numbers
- Each off-diagonal element is the negative of the other → The algebra is isomorphic to the complex numbers
Note: Each matrix of the third type, divided by the determinant, is a rotation matrix of the two-dimensional space; each matrix of the second type by its determinant (if it is different from 0 ) corresponds to a Lorentz transformation in a 1 1- dimensional Minkowski space.
Comments
- With the doubling method ( also known as the Cayley - Dickson process) can be generated whose dimension is twice as large as that of the starting number system new hypercomplex number systems.
- Every Clifford algebra is an associative hypercomplex number system.