[permalink] [id link]
* The quaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.
from
Wikipedia
Some Related Sentences
quaternions and form
* 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 ).
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 real
* 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.
The best known example is the ring of quaternions H. If we allow only rational instead of real coefficients in the constructions of the quaternions, we obtain another division ring.
* The real and rational quaternions ( q. v.
In 1846 Hamilton divided his quaternions into the sum of real and imaginary parts that he respectively called " scalar " and " vector ":
As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems.
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.
The multiplication of quaternions is not quite like the multiplication of real numbers, though.
In other words, the correct reasoning is the addition of two quaternions, one with zero vector / imaginary part, and another one with zero scalar / real part:
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.
Topologically, SO ( 3 ) is the real projective space RP < sup > 3 </ sup >, with fundamental group Z / 2, and only ( non-trivial ) covering space the hypersphere S < sup > 3 </ sup >, which is the group Spin ( 3 ), and represented by the unit quaternions.
It is a Lie group if K is the real or complex field or quaternions.
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:
This is a topological constraint – there is no covering map from the 3-torus to the 3-dimensional real projective space ; the only ( non-trivial ) covering map is from the 3-sphere, as in the use of quaternions.
* The unique simple formally real Jordan algebra, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions, is 27-dimensional.
Let R be the field of real numbers, C be the field of complex numbers, and H the quaternions.
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 ).
quaternions and algebra
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.
) 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 )
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.
* 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.
quaternions and with
In a three dimensional cartesian space a similar identification can be made with a subset of the quaternions.
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.
* The 3-sphere S < sup > 3 </ sup > forms a Lie group by identification with the set of quaternions of unit norm, called versors.
* 1843 – Sir William Rowan Hamilton comes up with the idea of quaternions, a non-commutative extension of complex 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 ).
This description as the quaternions of norm one, identifies the 3-sphere with the versors in the quaternion division ring.
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.
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 ).
quaternions and, with multiplication and conjugation defined exactly as for the quaternions:
This can be accomplished by choosing a curve such as the spherical linear interpolation in the quaternions, with one endpoint being the identity transformation 1 ( or some other initial rotation ) and the other being the intended final rotation.
The three-dimensional elliptic geometry makes use of the 3-sphere S < sup > 3 </ sup >, and these points are well-accessed with the versors in the theory of quaternions.
Hamilton solved this problem using the Icosian Calculus, an algebraic structure based on roots of unity with many similarities to the quaternions ( also invented by Hamilton ).
His earliest work dealt mainly with mathematical subjects, and especially with quaternions, which he was the leading exponent of after their originator, Hamilton.
* 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
0.174 seconds.