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* The quaternions form a 4-dimensional unitary associative algebra over the reals ( but not an algebra over the complex numbers, since if complex numbers are treated as a subset of the quaternions, complex numbers and quaternions do not commute ).
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quaternions and form
* The quaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.
In 1884 he recast Maxwell's mathematical analysis from its original cumbersome form ( they had already been recast as quaternions ) to its modern vector terminology, thereby reducing twelve of the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell's equations.
Inasmuch as quaternions consist of two independent complex numbers, they form a 4-dimensional vector space over the real numbers.
Inasmuch as octonions consist of two quaternions, the octonions form an 8-dimensional vector space over the real numbers.
The 24 Hurwitz quaternions of norm 1 form the 24-cell.
Nevertheless, the Weinberg form is consistent with Hyperbolic quaternions, a forerunner of Minkowski space.
The quaternions H form a 4 dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals ( see below ).
The quaternion can be related to the rotation vector form of the axis angle rotation by the exponential map over the quaternions,
* The quaternions form a noncommutative domain.
The hyperbolic quaternions of Alexander Macfarlane ( 1891 ) form a nonassociative ring that suggested the mathematical footing for spacetime theory that followed later.
quaternions and 4-dimensional
Although there are no division algebras in 3 dimensions, in 1843, the quaternions were defined and provided the now famous 4-dimensional example of an algebra over the real numbers, where one can not only multiply vectors, but also divide.
quaternions and unitary
The group Spin ( 3 ) is isomorphic to the special unitary group SU ( 2 ); it is also diffeomorphic to the unit 3-sphere S < sup > 3 </ sup > and can be understood as the group of unit quaternions ( i. e. those with absolute value 1 ).
* The unit 3-sphere centered at 0 in the quaternions H is a Lie group ( isomorphic to the special unitary group ) whose tangent space at 1 can be identified with the space of purely imaginary quaternions, The exponential map for this Lie group is given by
The group Spin ( 3 ) is isomorphic to the special unitary group SU ( 2 ); it is also diffeomorphic to the unit 3-sphere S < sup > 3 </ sup > and can be understood as the group of unit quaternions ( i. e. those with absolute value 1 ).
Section IV: Versors and unitary vectors in the system of quaternions.
quaternions and associative
* Addition and multiplication of complex numbers and quaternions is associative.
Yet Clifford is now best remembered for his eponymous Clifford algebras, a type of associative algebra that generalizes the complex numbers and William Rowan Hamilton's quaternions and biquaternions.
Since quaternion multiplication is associative, and real numbers commute with other quaternions, the norm of a product of quaternions equals the product of the norms:
quaternions and algebra
* Every real Banach algebra which is a division algebra is isomorphic to the reals, the complexes, or the quaternions.
* Every unital real Banach algebra with no zero divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions.
With the identification of the even-graded elements with the algebra H of quaternions, as in the case of two-dimensions the only representation of the algebra of even-graded elements is on itself.
Whitehead's early work sought to unify quaternions ( due to Hamilton ), Grassmann's Ausdehnungslehre, and Boole's algebra of logic.
The term " sedenion " is also used for other 16-dimensional algebraic structures, such as a tensor product of 2 copies of the quaternions, or the algebra of 4 by 4 matrices over the reals, or that studied by.
There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H. The octonions are the largest such algebra, with eight dimensions, double the number of the quaternions from which they are an extension.
) In 1878, William Kingdon Clifford joined this exterior algebra to William Rowan Hamilton's quaternions by replacing Grassmann's rule e < sub > p </ sub > e < sub > p </ sub >
Unlike the complex numbers, the quaternions are an example of a non-commutative algebra: for instance, ( 0, 1, 0, 0 ) · ( 0, 0, 1, 0 )
* More generally, the conjugation involution in any Cayley – Dickson algebra such as the complex numbers, quaternions and octonions, if a blind eye is turned on the nonassociativity of the latter.
A 3D graphics programmer must have a firm grasp of advanced mathematical concepts such as vector and matrix math, quaternions and linear algebra.
* The unique simple formally real Jordan algebra, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions, is 27-dimensional.
quaternions and over
For square matrices with entries in a non-commutative ring, for instance the quaternions, there is no unique definition for the determinant, and no definition that has all the usual properties of determinants over commutative rings.
Frobenius theorem: The only finite-dimensional division algebras over the reals are the reals themselves, the complex numbers, and the quaternions.
However, for matrices over a more general ring that need not be commutative, such as the quaternions, they may not be equal.
The quaternions were soon followed by several other hypercomplex number systems, which were the early examples of algebras over a field.
The finite-dimensional division algebras with center R ( that means the dimension over R is finite ) are the real numbers and the quaternions by a theorem of Frobenius, while any matrix ring over the reals or quaternions – M ( n, R ) or M ( n, H ) – is a CSA over the reals, but not a division algebra ( if ).
In particular, the only simple rings that are a finite-dimensional vector space over the real numbers are rings of matrices over either the real numbers, the complex numbers, or the quaternions.
Like the quaternions, it has dimension 4 over R, but unlike the quaternions, it has zero divisors, as can be seen from the following product of the matrix units: E < sub > 11 </ sub > E < sub > 21 </ sub > = 0, hence it is not a division ring.
quaternions and reals
Each of these spaces forms an H-space by viewing it as the subset of norm-one elements of the reals, complexes, quaternions, and octonions, respectively, and using the multiplication operations from these algebras.
In addition to examples from differential topology ( such as characteristic classes ), Arnold considers the three Platonic symmetries ( tetrahedral, octahedral, icosahedral ) as corresponding to the reals, complexes, and quaternions, which then connects with McKay's more algebraic correspondences, below.
* Projective spaces over the reals, complexes, or quaternions are compact manifolds.
It is a theorem of Frobenius that there are only two real quaternion algebras: 2 × 2 matrices over the reals and Hamilton's real quaternions.
quaternions and complex
For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces.
* 1843 – Sir William Rowan Hamilton comes up with the idea of quaternions, a non-commutative extension of complex numbers.
The immediate predecessor of vectors were quaternions, devised by William Rowan Hamilton in 1843 as a generalization of complex numbers.
Whereas complex numbers have one number whose square is negative one, quaternions have three independent such numbers.
It is often convenient to regard R < sup > 4 </ sup > as the space with 2 complex dimensions ( C < sup > 2 </ sup >) or the quaternions ( H ).
These transformations may also be written as biquaternions ( quaternions with complex elements ), where the elements are related to the Jones matrix in the same way that the Stokes parameters are related to the coherency matrix.
As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems.
Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used.
If the and are real numbers, a more elegant proof is available: the identity expresses the fact that the absolute value of the product of two quaternions is equal to the product of their absolute values, in the same way that the Brahmagupta – Fibonacci two-square identity does for complex numbers.
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