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Composition Algebra
In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involution called a conjugation: x \mapsto x^*. The quadratic form N(x) = x x^* is called the norm of the algebra. A composition algebra (''A'', ∗, ''N'') is either a division algebra or a split algebra, depending on the existence of a non-zero ''v'' in ''A'' such that ''N''(''v'') = 0, called a null vector. When ''x'' is ''not'' a null vector, the multiplicative inverse of ''x'' is When there is a non-zero null vector, ''N'' is an isotropic quadratic form, and "the algebra splits". Structure theorem Every unital composition algebra over a field can be obtained by repeated application of the Cayley–Dickson construction starting from (if the characteristic of is different from ) or a 2-dimensional composition subalgebra (if ). The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Quadratic Field Extension
In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ''L''. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry. Subfield A subfield K of a field L is a subset K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains the multiplicative identity 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K. As , the latter definition implies K and L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Split Algebra
In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector. A quadratic space which has a null vector is called a pseudo-Euclidean space. The term ''isotropic vector v'' when ''q''(''v'') = 0 has been used in quadratic spaces, and anisotropic space for a quadratic space without null vectors. A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces ''A'' and ''B'', , where ''q'' is positive-definite on ''A'' and negative-definite on ''B''. The null cone, or isotropic cone, of ''X'' consists of the union of balanced spheres: \bigcup_ \. The null cone is also the union of the isotropic lines through the origin. Split algebras A comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Square (algebra)
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations ''x''^2 ( caret) or ''x''**2 may be used in place of ''x''2. The adjective which corresponds to squaring is '' quadratic''. The square of an integer may also be called a '' square number'' or a ''perfect square''. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial is the quadratic polynomial . One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Pauli Algebra
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in connection with isospin symmetries. \begin \sigma_1 = \sigma_x &= \begin 0&1\\ 1&0 \end, \\ \sigma_2 = \sigma_y &= \begin 0& -i \\ i&0 \end, \\ \sigma_3 = \sigma_z &= \begin 1&0\\ 0&-1 \end. \\ \end These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation, which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left). Each Pauli matrix is Hermitian, and together with the id ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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William Rowan Hamilton
Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathematical equations are considered fundamental to modern theoretical physics, particularly Hamiltonian mechanics, his reformulation of Lagrangian mechanics. His career included the analysis of geometrical optics, Fourier analysis, and quaternions, the last of which made him one of the founders of modern linear algebra. Hamilton was Andrews Professor of Astronomy at Trinity College Dublin. He was also the third director of Dunsink Observatory from 1827 to 1865. The Hamilton Institute at Maynooth University is named after him. Early life Hamilton was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819), who lived in Dublin at 29 Dominick Street, Dublin, Dominick Street, later renumbered to 36. Ham ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Bioctonion
In mathematics, a bioctonion, or complex octonion, is a pair (''p,q'') where ''p'' and ''q'' are biquaternions. The product of two bioctonions is defined using biquaternion multiplication and the biconjugate p → p*: :(p,q)(r,s) = (pr - s^* q,\ sp + q r^*). The bioctonion ''z'' = (''p,q'') has conjugate ''z''* = (''p''*, – ''q''). Then norm ''N''(''z'') of bioctonion ''z'' is ''z z''* = ''p p''* + ''q q''*, which is a complex quadratic form with eight terms. The bioctonion algebra is sometimes introduced as simply the complexification of real octonions, but in abstract algebra it is the result of the Cayley–Dickson construction that begins with the field of complex numbers, the trivial involution, and quadratic form z2. The algebra of bioctonions is an example of an octonion algebra. For any pair of bioctonions ''y'' and ''z'', : N(y z) = N(y) N(z), showing that ''N'' is a quadratic form admitting composition, and hence the bioctonions form a composition algebra. Guy Roo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Matrix Ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication. The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'') (alternative notations: Mat''n''(''R'') and ). Some sets of infinite matrices form infinite matrix rings. A subring of a matrix ring is again a matrix ring. Over a rng, one can form matrix rngs. When ''R'' is a commutative ring, the matrix ring M''n''(''R'') is an associative algebra over ''R'', and may be called a matrix algebra. In this setting, if ''M'' is a matrix and ''r'' is in ''R'', then the matrix ''rM'' is the matrix ''M'' with each of its entries multiplied by ''r''. Examples * The set of all square matrices over ''R'', denoted M''n''(''R''). This is sometimes called the "full ring of ''n''-by-''n'' matrices". * The set of all upper triangular matrices over ''R''. * The set of all lower triangular matrices over ''R''. * The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: * Biquaternions when the coefficients are complex numbers. * Split-biquaternions when the coefficients are split-complex numbers. * Dual quaternions when the coefficients are dual numbers. This article is about the ''ordinary biquaternions'' named by William Rowan Hamilton in 1844. Some of the more prominent proponents of these biquaternions include Alexander Macfarlane, Arthur W. Conway, Ludwik Silberstein, and Cornelius Lanczos. As developed below, the unit quasi-sphere of the biquaternions provides a representation of the Lorentz group, which is the foundation of special relativity. The algebra of biquaternions can be considered as a tensor product of algebras, tensor product , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Bicomplex Number
In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^* = (w, -z), and the product of two bicomplex numbers as : (u,v)(w,z) = (u w - v z, u z + v w). Then the bicomplex norm is given by : (w,z)^* (w,z) = (w, -z)(w,z) = (w^2 + z^2, 0), a quadratic form in the first component. The bicomplex numbers form a commutative algebra over C of dimension two that is isomorphic to the direct sum of algebras . The product of two bicomplex numbers yields a quadratic form value that is the product of the individual quadratic forms of the numbers: a verification of this property of the quadratic form of a product refers to the Brahmagupta–Fibonacci identity. This property of the quadratic form of a bicomplex number indicates that these numbers form a composition algebra. In fact, bicomplex numbers arise at the binarion level of the Cayley–Dickson construction based on \mathbb with no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |