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Split Quaternion
In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers. After introduction in the 20th century of coordinate-free definitions of rings and algebras, it was proved that the algebra of split-quaternions is isomorphic to the ring of the real matrices. So the study of split-quaternions can be reduced to the study of real matrices, and this may explain why there are few mentions of split-quaternions in the mathematical literature of the 20th and 21st centuries. Definition The ''split-quaternions'' are the linear combinations (with real coefficients) of four basis elements that satisfy the following product rules: : , : , : , : . By associativity, these relations imply : , : , and also . So, the split-quaternions form a real vector space of dimension four with as a basis. They form also a noncommutative ring, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis (linear algebra), basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible matrix, invertible and the corresponding linear map is an linear isomorphism, isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norm (mathematics)
In mathematics, a norm is a function (mathematics), function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the Origin (mathematics), origin: it Equivariant map, commutes with scaling, obeys a form of the triangle inequality, and zero is only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the #Euclidean norm, Euclidean norm, the #p-norm, 2-norm, or, sometimes, the magnitude or length of the vector. This norm can be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a ''seminormed vector space''. The term pseudonorm has been used for several related meaning ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropic Quadratic Form
In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise it is a definite quadratic form. More explicitly, if ''q'' is a quadratic form on a vector space ''V'' over ''F'', then a non-zero vector ''v'' in ''V'' is said to be isotropic if . A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form. Suppose that is quadratic space and ''W'' is a subspace of ''V''. Then ''W'' is called an isotropic subspace of ''V'' if ''some'' vector in it is isotropic, a totally isotropic subspace if ''all'' vectors in it are isotropic, and a definite subspace if it does not contain ''any'' (non-zero) isotropic vectors. The of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces. Over the real numbers, more generally in the case where ''F'' is a real closed field (so that the signature is defined), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cofactor Matrix
In linear algebra, a minor of a matrix is the determinant of some smaller square matrix generated from by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which are useful for computing both the determinant and inverse of square matrices. The requirement that the square matrix be smaller than the original matrix is often omitted in the definition. Definition and illustration First minors If is a square matrix, then the ''minor'' of the entry in the -th row and -th column (also called the ''minor'', or a ''first minor'') is the determinant of the submatrix formed by deleting the -th row and -th column. This number is often denoted . The ''cofactor'' is obtained by multiplying the minor by . To illustrate these definitions, consider the following matrix, \begin 1 & 4 & 7 \\ 3 & 0 & 5 \\ -1 & 9 & 11 \\ \end To compute the minor and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotent Element (ring Theory)
Idempotence (, ) is the property of certain operation (mathematics), operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projector (linear algebra), projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency). The term was introduced by American mathematician Benjamin Peirce in 1870 in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from + ''wikt:potence, potence'' (same + power). Definition An element x of a set S equipped with a binary operator \cdot is said to be ''idempotent'' under \cdot if : . The ''binary operation'' \cdot is said to be ''idempotent'' if : . Examples * In the monoid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotent
In mathematics, an element x of a ring (mathematics), ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term, along with its sister Idempotent (ring theory), idempotent, was introduced by Benjamin Peirce in the context of his work on the classification of algebras. Examples *This definition can be applied in particular to square matrix, square matrices. The matrix :: A = \begin 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end :is nilpotent because A^3=0. See nilpotent matrix for more. * In the factor ring \Z/9\Z, the equivalence class of 3 is nilpotent because 32 is Congruence relation, congruent to 0 Modular arithmetic, modulo 9. * Assume that two elements a and b in a ring R satisfy ab=0. Then the element c=ba is nilpotent as \beginc^2&=(ba)^2\\ &=b(ab)a\\ &=0.\\ \end An example with matrices (for ''a'', ''b''):A = \begin 0 & 1\\ 0 & 1 \end, \;\; B =\begin 0 & 1\\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero divisor if there exists a nonzero in such that . This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero such that may be different from the nonzero such that ). If the ring is commutative, then the left and right zero divisors are the same. An element of a ring that is not a left zero divisor (respectively, not a right zero divisor) is called left regular or left cancellable (respectively, right regular or right cancellable). An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called regular or cancellabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension (vector Space)
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say V is if the dimension of V is wiktionary:finite, finite, and if its dimension is infinity, infinite. The dimension of the vector space V over the field F can be written as \dim_F(V) or as [V : F], read "dimension of V over F". When F can be inferred from context, \dim(V) is typically written. Examples The vector space \R^3 has \left\ as a standard basis, and therefore \dim_(\R^3) = 3. More generally, \dim_(\R^n) = n, and even more generally, \dim_(F^n) = n for any Field (mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quaternions is often denoted by (for ''Hamilton''), or in blackboard bold by \mathbb H. Quaternions are not a field, because multiplication of quaternions is not, in general, commutative. Quaternions provide a definition of the quotient of two vectors in a three-dimensional space. Quaternions are generally represented in the form a + b\,\mathbf i + c\,\mathbf j +d\,\mathbf k, where the coefficients , , , are real numbers, and , are the ''basis vectors'' or ''basis elements''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, robotics, magnetic resonance i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
