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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] |
Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He postulated what is now known as the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial, and verified it for matrices of order 2 and 3. He was the first to define the concept of an abstract group, a set with a binary operation satisfying certain laws, as opposed to Évariste Galois' concept of permutation groups. In group theory, Cayley tables, Cayley graphs, and Cayley's theorem are named in his honour, as well as Cayley's formula in combinatorics. Early life Arthur Cayley was born in Richmond, London, England, on 16 August 1821. His father, Henry Cayley, was a distant cousin of George Cayley, the aeronautics engineer innovator, and descended from an ancient Yorkshire family. He settled i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monic Polynomial
In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. That is to say, a monic polynomial is one that can be written as :x^n+c_x^+\cdots+c_2x^2+c_1x+c_0, with n \geq 0. Uses Monic polynomials are widely used in algebra and number theory, since they produce many simplifications and they avoid divisions and denominators. Here are some examples. Every polynomial is associated to a unique monic polynomial. In particular, the unique factorization property of polynomials can be stated as: ''Every polynomial can be uniquely factorized as the product of its leading coefficient and a product of monic irreducible polynomials.'' Vieta's formulas are simpler in the case of monic polynomials: ''The th elementary symmetric function of the roots of a monic polynomial of degree equals (-1)^ic_, where c_ is the coefficient of the th po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Expression
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (mathematics), ring formed from the set (mathematics), set of polynomials in one or more indeterminate (variable), indeterminates (traditionally also called variable (mathematics), variables) with coefficients in another ring (mathematics), ring, often a field (mathematics), field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the Integer#Algebraic_properties, integers. Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, formal power series, rings of formal power series, Ore polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invertible Matrix
In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix. Invertible matrices are the same size as their inverse. Definition An -by- square matrix is called invertible if there exists an -by- square matrix such that\mathbf = \mathbf = \mathbf_n ,where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. Over a field, a square matrix that is ''not'' invertible is called singular or deg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. He was also the first to introduce the notion of rational approximations of functions (nowadays known as Padé approximants), and gave the first full proof for the Cayley–Hamilton theorem. He also lent his name to certain differential-geometric objects in modern mathematical physics, known as Frobenius manifolds. Biography Ferdinand Georg Frobenius was born on 26 October 1849 in Charlottenburg, a suburb of Berlin, from parents Christian Ferdinand Frobenius, a Protestant parson, and Christine Elizabeth Friedrich. He entered the Joachimsthal Gymnasium in 1860 when he was nearly el ... [...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] |
Polynomial Division
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones. Sometimes using a shorthand version called synthetic division is faster, with less writing and fewer calculations. Another abbreviated method is polynomial short division (Blomqvist's method). Polynomial long division is an algorithm that implements the Euclidean division of polynomials, which starting from two polynomials ''A'' (the ''dividend'') and ''B'' (the ''divisor'') produces, if ''B'' is not zero, a ''quotient'' ''Q'' and a ''remainder'' ''R'' such that :''A'' = ''BQ'' + ''R'', and either ''R'' = 0 or the degree of ''R'' is lower than the degree of ''B''. These conditions uniquely define ''Q'' and ''R'', which means that ''Q'' and ''R'' do not depend on th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Polynomial (linear Algebra)
Minimal may refer to: * Minimal (music genre), art music that employs limited or minimal musical materials * "Minimal" (song), 2006 song by Pet Shop Boys * Minimal (supermarket) or miniMAL, a former supermarket chain in Germany and Poland * Minimal (''Dungeons & Dragons''), a creature of magically reduced size in the game ''Dungeons & Dragons'' * Minimal (chocolate), a bean to bar chocolate store in Japan, featured in '' Kantaro: The Sweet Tooth Salaryman'' * Minimal (clothing), an Indonesia clothing-retail company that worked with fashion model Ayu Gani * MINIMAL (restaurant), high end restaurant in Taichung Taichung (, Wade–Giles: '), officially Taichung City, is a special municipality (Taiwan), special municipality in central Taiwan. Taichung is Taiwan's second-largest city, with more than 2.85 million residents, making it the largest city in Ce ..., Taiwan See also * * Minimalism (other) * Maximal (other) * Minimisation (other) * Minimal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be any expression of the form ''ax'' + ''by'', where ''a'' and ''b'' are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field (mathematics), field, with some generalizations given at the end of the article. Definition Let ''V'' be a vector space over the field ''K''. As usual, we call elements of ''V'' ''vector space, vectors'' and call elements of ''K'' ''scalar (mathematics), scalars''. If v1,...,v''n'' are vectors and ''a''1,...,''a''''n'' are scalars, then the ''linear combination of those vectors with those scalars as coefficients'' is :a_1 \mathbf v_1 + a_2 \mathbf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formal system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annihilating Polynomial
A polynomial ''P'' is ''annihilating'' or called an annihilating polynomial in linear algebra and operator theory if the polynomial considered as a function of the linear operator or a matrix ''A'' evaluates to zero, i.e., is such that ''P(A) = 0''. Note that all characteristic polynomials and minimal polynomials of ''A'' are annihilating polynomials. In fact, every annihilating polynomial is the multiple of the minimal polynomial of an operator ''A''.Hoffman, K., Kunze, R., "Linear Algebra", 2nd ed., 1971, Prentice-Hall. (Definition on page 191 of section 6.3) See also * Cayley–Hamilton theorem * Minimal polynomial (linear algebra) Minimal may refer to: * Minimal (music genre), art music that employs limited or minimal musical materials * "Minimal" (song), 2006 song by Pet Shop Boys * Minimal (supermarket) or miniMAL, a former supermarket chain in Germany and Poland * Mini ... References {{reflist Matrix theory Polynomials Linear algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
