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Central Polynomial
In algebra, a central polynomial for ''n''-by-''n'' matrices is a polynomial in non-commuting variables that is non-constant but yields a scalar matrix whenever it is evaluated at ''n''-by-''n'' matrices. That such polynomials exist for any square matrices was discovered in 1970 independently by Formanek and Razmyslov. The term "central" is because the evaluation of a central polynomial has the image lying in the center of the matrix ring over any commutative ring. The notion has an application to the theory of polynomial identity rings. Example: (xy - yx)^2 is a central polynomial for 2-by-2-matrices. Indeed, by the Cayley–Hamilton theorem, one has that (xy - yx)^2 = -\det(xy - yx)I for any 2-by-2-matrices ''x'' and ''y''. See also *Generic matrix ring In algebra, a generic matrix ring is a sort of a universal matrix ring. Definition We denote by F_n a generic matrix ring of size ''n'' with variables X_1, \dots X_m. It is characterized by the universal property: given a ... [...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] |
Matrix (mathematics)
In mathematics, a matrix (: matrices) is a rectangle, rectangular array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end 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 . Matrices are commonly used in linear algebra, where they represent linear maps. In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotation (mathematics), rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problem (mathematics education), word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Matrix
In mathematics, a square matrix is a Matrix (mathematics), matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear transformations, such as Shear mapping, shearing or Rotation (mathematics), rotation. For example, if R is a square matrix representing a rotation (rotation matrix) and \mathbf is a column vector describing the Position (vector), position of a point in space, the product R\mathbf yields another column vector describing the position of that point after that rotation. If \mathbf is a row vector, the same transformation can be obtained using where R^ is the transpose of Main diagonal The entries a_ () form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edward W
Edward is an English language, English male name. It is derived from the Old English, Anglo-Saxon name ''Ēadweard'', composed of the elements ''wikt:ead#Old English, ēad'' "wealth, fortunate; prosperous" and ''wikt:weard#Old English, weard'' "guardian, protector”. History The name Edward was very popular in Anglo-Saxon England, but the rule of the House of Normandy, Norman and House of Plantagenet, Plantagenet dynasties had effectively ended its use amongst the upper classes. The popularity of the name was revived when Henry III of England, Henry III named his firstborn son, the future Edward I of England, Edward I, as part of his efforts to promote a cult around Edward the Confessor, for whom Henry had a deep admiration. Variant forms The name has been adopted in the Iberian Peninsula#Modern Iberia, Iberian peninsula since the 15th century, due to Edward, King of Portugal, whose mother was English. The Spanish/Portuguese forms of the name are Eduardo and Duarte (name), Duart ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Center (ring Theory)
In algebra, the center of a ring ''R'' is the subring consisting of the elements ''x'' such that for all elements ''y'' in ''R''. It is a commutative ring and is denoted as Z(''R''); 'Z' stands for the German word ''Zentrum'', meaning "center". If ''R'' is a ring, then ''R'' is an associative algebra over its center. Conversely, if ''R'' is an associative algebra over a commutative subring ''S'', then ''S'' is a subring of the center of ''R'', and if ''S'' happens to be the center of ''R'', then the algebra ''R'' is called a central algebra. Examples * The center of a commutative ring ''R'' is ''R'' itself. * The center of a skew-field is a field. * The center of the (full) matrix ring with entries in a commutative ring ''R'' consists of ''R''-scalar multiples of the identity matrix. * Let ''F'' be a field extension of a field ''k'', and ''R'' an algebra over ''k''. Then . * The center of the universal enveloping algebra of a Lie algebra In mathematics, a Lie algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Commutative Ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Commutative rings appear in the following chain of subclass (set theory), class inclusions: Definition and first examples Definition A ''ring'' is a Set (mathematics), set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Identity Ring
In ring theory, a branch of mathematics, a ring ''R'' is a polynomial identity ring if there is, for some ''N'' > 0, an element ''P'' ≠ 0 of the free algebra, Z, over the ring of integers in ''N'' variables ''X''1, ''X''2, ..., ''X''''N'' such that :P(r_1, r_2, \ldots, r_N) = 0 for all ''N''-tuples ''r''1, ''r''2, ..., ''r''''N'' taken from ''R''. Strictly the ''X''''i'' here are "non-commuting indeterminates", and so "polynomial identity" is a slight abuse of language, since "polynomial" here stands for what is usually called a "non-commutative polynomial". The abbreviation PI-ring is common. More generally, the free algebra over any ring ''S'' may be used, and gives the concept of PI-algebra. If the degree of the polynomial ''P'' is defined in the usual way, the polynomial ''P'' is called monic if at least one of its terms of highest degree has coefficient equal to 1. Every commutative ring is a PI-ring, satisfying the polynomial identity ''XY'' − ''YX'' = 0. Therefore, PI ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley–Hamilton Theorem
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies its own characteristic equation. The characteristic polynomial of an matrix is defined as p_A(\lambda)=\det(\lambda I_n-A), where is the determinant operation, is a variable scalar element of the base ring, and is the identity matrix. Since each entry of the matrix (\lambda I_n-A) is either constant or linear in , the determinant of (\lambda I_n-A) is a degree- monic polynomial in , so it can be written as p_A(\lambda) = \lambda^n + c_\lambda^ + \cdots + c_1\lambda + c_0. By replacing the scalar variable with the matrix , one can define an analogous matrix polynomial expression, p_A(A) = A^n + c_A^ + \cdots + c_1A + c_0I_n. (Here, A is the given matrix—not a variable, unlike \lambda—so p_A(A) is a constant rather than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generic Matrix Ring
In algebra, a generic matrix ring is a sort of a universal matrix ring. Definition We denote by F_n a generic matrix ring of size ''n'' with variables X_1, \dots X_m. It is characterized by the universal property: given a commutative ring ''R'' and ''n''-by-''n'' matrices A_1, \dots, A_m over ''R'', any mapping X_i \mapsto A_i extends to the ring homomorphism (called evaluation) F_n \to M_n(R). Explicitly, given a field ''k'', it is the subalgebra F_n of the matrix ring M_n(k X_l)_ \mid 1 \le l \le m,\ 1 \le i, j \le n generated by ''n''-by-''n'' matrices X_1, \dots, X_m, where (X_l)_ are matrix entries and commute by definition. For example, if ''m'' = 1 then F_1 is a polynomial ring in one variable. For example, a central polynomial is an element of the ring F_n that will map to a central element under an evaluation. (In fact, it is in the invariant ring k X_l)_ since it is central and invariant.) By definition, F_n is a quotient of the free ring k\langle t_1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
