Cayley Algebra
Cayley algebra is the branch of the non-commutative algebras dealing with matrices. English mathematician Arthur Cayley developed it during 1840 to 1890. It is the only non-associative division algebra with real scalars. Division algebra is a type of algebra in which every nonzero element has a multiplicative inverse but where multiplication is non-commutative, that is x * y y * x.
Arthur Cayley, considered a British mathematician although he practiced law the first 14 years of his professional career, was part of the movement in the 19th century by British mathematicians to study algebras. During these studies of various sorts of mathematical objects Cayley turned his attention to matrices and assorted operations that could be performed on them. This was a time when the scope of algebra was expanded and not limited to ordinary systems of numbers alone and one of the most important developments was the formulation of non-commutative algebras, those in which the operation of multiplication is not required to be commutative. William Rowan Hamilton was the first to develop such an algebra in 1843 with quaternions. Quaternions are an important example of a four-dimensional vector space that was employed by Einstein to develop his theory of special relativity. Two years before, Cayley had begun the first English contribution to the theory of determinants. During the late 1840s Cayley went to a lecture Hamilton was giving on quaternions. He continued his development of matrix algebra and in 1853 Cayley published an article giving the inverse of a matrix.
In the late 1850s Cayley formally introduced matrix algebras when he published Memoir on the theory of matrices in 1858 and noticed that quaternions could be represented by matrices. He went on to show that the coefficient arrays studied previously for quadratic forms and for linear transformations are special cases of his general concepts. In his book Cayley developed methods for carrying out addition, multiplication, scalar multiplication and inverses on matrices. He gave specifics of relating the construction of the inverse of a matrix in terms of the determinant of that matrix. He developed the theory of algebraic invariance and his work on matrices was the basis for quantum mechanics developed by Werner Heisenberg in 1925.
Cayley algebra is a non-associative algebra in that it does not satisfy a(bc) = (ab)c. It is a division algebra in which there is an eight-square identity and the elements are called Cayley numbers or octonions. Cayley numbers are typically of the form: a + bi0 + ci1 + di2 + ei3 + fi4 + gi5 + hi6. Each of the triples (i0, i1, i3), (i1, i2, i4), (i2, i3, i5), (i3, i4, i6), (i4, i5, i0), (i5, i6, i1), (i6, i0, i2) behaves like the quaternions (i, j, k). As mentioned before quaternions are a set of symbols that have the form: a + bi + cj + dk, where a, b, c, and d are real numbers, and they multiply using the rules: i2 = j2 = k2 = -1 and ij = k. These numbers used in Cayley algebra are not associative. They have been employed in the study of seven and eight-dimensional space. Cayley algebra has particularly important used in physical applications.
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