In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.
In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere (or 2-sphere, a two-dimensional surface), the boundary of a ball in four dimensions is a 3-sphere (an object with three dimensions). A 3-sphere is an example of a 3-manifold and an n-sphere.
Blackboard bold is a style of writing bold symbols on a blackboard by doubling certain strokes, commonly used in mathematical lectures, and the derived style of typeface used in printed mathematical texts. The style is most commonly used to represent the number sets (natural numbers), (integers), (rational numbers), (real numbers), and (complex numbers). To imitate a bold typeface on a typewriter, a character can be typed over itself (called double-striking); symbols thus produced are called double-struck, and this name is sometimes adopted for blackboard bold symbols, for instance in Unicode glyph names.
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and .
In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers, usually represented by the capital letter S, boldface S or blackboard bold . They are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic to a subalgebra of the sedenions. Unlike the octonions, the sedenions are not an alternative algebra. Applying the Cayley–Dickson construction to the sedenions yields a 32-dimensional algebra, sometimes called the 32-ions or trigintaduonions.
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface O or blackboard bold . Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.
In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, added to the real and complex numbers.
In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions.
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries. 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.
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol . Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming.
