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Matrix Exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let be an real or complex matrix. The exponential of , denoted by or , is the matrix given by the power series e^X = \sum_^\infty \frac X^k where X^0 is defined to be the identity matrix I with the same dimensions as X. The above series always converges, so the exponential of is well-defined. If is a 1×1 matrix the matrix exponential of is a 1×1 matrix whose single element is the ordinary exponential of the single element of . Properties Elementary properties Let and be complex matrices and let and be arbitrary complex numbers. We denote the identity matrix by and the zero matrix by 0. The matrix exponential satisfies the foll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequalities For Exponentials Of Hermitian Matrices
Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * Spatial inequality, the unequal distribution of income and resources across geographical regions * Income inequality metrics, used to measure income and economic inequality among participants in a particular economy * International inequality, economic differences between countries Healthcare * Health equity, the study of differences in the quality of health and healthcare across different populations Mathematics * Inequality (mathematics), a relation between two values when they are different Social sciences * Educational inequality, the unequal distribution of academic resources to socially excluded communities * Gender inequality, unequal treatment or perceptions of individuals due to their gender * Participation inequality, the phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Identity
In mathematics, a trace identity is any equation involving the trace of a matrix. Properties Trace identities are invariant under simultaneous conjugation. Uses They are frequently used in the invariant theory of n \times n matrices to find the generators and relations of the ring of invariants, and therefore are useful in answering questions similar to that posed by Hilbert's fourteenth problem. Examples * The Cayley–Hamilton theorem says that every square matrix satisfies its own characteristic polynomial. This also implies that all square matrices satisfy \operatorname\left(A^n\right) - c_\operatorname(A) \operatorname\left(A^\right) + \cdots + (-1)^n n \det(A) = 0\, where the coefficients c_i are given by the elementary symmetric polynomials of the eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi's Formula
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix ''A'' in terms of the adjugate of ''A'' and the derivative of ''A''., Part Three, Section 8.3 If is a differentiable map from the real numbers to matrices, then : \frac \det A(t) = \operatorname \left (\operatorname(A(t)) \, \frac\right ) = \left(\det A(t) \right) \cdot \operatorname \left (A(t)^ \cdot \, \frac\right ) where is the trace of the matrix . (The latter equality only holds if ''A''(''t'') is invertible.) As a special case, : = \operatorname(A)_. Equivalently, if stands for the differential of , the general formula is : d \det (A) = \operatorname (\operatorname(A) \, dA). The formula is named after the mathematician Carl Gustav Jacob Jacobi. Derivation Via Matrix Computation We first prove a preliminary lemma: Lemma. Let ''A'' and ''B'' be a pair of square matrices of the same dimension ''n''. Then :\sum_i \sum_j A_ B_ = \operatorname (A^ B). ''Proof.'' The produc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnus Series
In mathematics and physics, the Magnus expansion, named after Wilhelm Magnus (1907–1990), provides an exponential representation of the solution of a first-order homogeneous linear differential equation for a linear operator. In particular, it furnishes the fundamental matrix of a system of linear ordinary differential equations of order with varying coefficients. The exponent is aggregated as an infinite series, whose terms involve multiple integrals and nested commutators. The deterministic case Magnus approach and its interpretation Given the coefficient matrix , one wishes to solve the initial-value problem associated with the linear ordinary differential equation : Y'(t) = A(t) Y(t), \quad Y(t_0) = Y_0 for the unknown -dimensional vector function . When ''n'' = 1, the solution simply reads : Y(t) = \exp \left( \int_^t A(s)\,ds \right) Y_0. This is still valid for ''n'' > 1 if the matrix satisfies for any pair of values of ''t'', ''t''1 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applications Application may refer to: Mathematics and computing * Application software, computer software designed to help the user to perform specific tasks ** Application layer, an abstraction layer that specifies protocols and interface methods used in a communications network * Function application, in mathematics and computer science Processes and documents * Application for employment, a form or forms that an individual seeking employment must fill out * College application, the process by which prospective students apply for entry into a college or university * Patent application A patent application is a request pending at a patent office for the grant of a patent for an invention described in the patent specification and a set of one or more claims stated in a formal document, including necessary official forms and rel ..., a document filed at a patent office to support the grant of a patent Other uses * Application (virtue), a chara |
