In mathematics, a quaternion algebra over a field L is a particular kind of central simple algebraA over L, namely such an algebra that has dimension 4, and therefore becomes the 2×2 matrix algebra over some field extension of L, by extending scalars. The classical quaternions are the case of L the real number field, and A is uniquely defined up to isomorphism by the condition that it is such a quaternion algebra that is not the 2×2 real matrix algebra. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java... In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K. It is an abelian group with elements isomorphism classes of division algebras over K, such that the center is exactly K. The group is named for the algebraist Richard Brauer. ... ... In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... Please refer to Real vs. ...
Quaternion algebra therefore means something other than the algebra of quaternions. In fact it is shown that any quaternion algebra is a slightly twisted form of the familiar quaternions, with a basic 1, i, j, k such that ij = k and adjusted so that for example
i2 = a
etc. where a need not be −1.
Quaternion algebras are applied in number theory, particularly to quadratic forms. Traditionally, number theory is the branch of pure mathematics concerned with the properties of integers. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...
The word "quaternion " properly means " a set of four." In employing such a word to denote a new mathematical method, Sir W. Hamilton was probably influenced by the recollection of its Greek equivalent, the Pythagorean Tetractys (TerpaKrt, the number four), the mystic source of all things.
Quaternions (as a mathematical method) is an extension, or improvement, of Cartesian geometry, in which the artifices of co-ordinate axes, andc., are got rid of, all directions in space being treated on precisely the same terms.
The evolution of quaternions belongs in part to each of two weighty branches of mathematical history - the interpretation of the imaginary (or impossible) quantity of common algebra, and the Cartesian application of algebra to geometry.
The conjugate of a quaternion corresponds to the conjugate transpose of the matrix.
In this representation, the conjugate of a quaternion corresponds to the transpose of the matrix.
Quaternions are often used in computer graphics (and associated geometric analysis) to represent rotations (see quaternions and spatial rotation) and orientations of objects in three-dimensional space.
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