Identity matrixIn linear algebra, the identity matrix of size n{\displaystyle n} is the n×n{\displaystyle n\times n}square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties; for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1.
Exponential functionMathematical function, denoted exp(x) or e^xIn mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted ex{\displaystyle e^{x}} or expx{\displaystyle \exp x}; the latter is preferred when the argument x{\displaystyle x} is a complicated expression.
Exponential map (Lie theory)Map from a Lie algebra to its Lie group Lie groups and Lie algebras Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8 Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing formIndex Simple Lie algebra Loop algebra Affine Lie algebra Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root systemWeyl group Real formComplexification Split Lie algebra Compact Lie algebra Representation theory Lie group representation Lie algebra representation Representation theory of semisimple Lie algebras Representations of classical groups Theorem of the highest weight Borel–Weil–Bott theorem Lie groups in physics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel Glossary Table of Lie groupsvte In the theory of Lie groups, the exponential map is a map from the Lie algebra g {\displaystyle {\mathfrak {g}}} of a Lie group G {\displaystyle G} to the group, which allows one to recapture the local group structure from the Lie algebra.
Square matrixIn mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n{\displaystyle n}. 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 shearing or rotation.
Complex numberNumber with a real and an imaginary partA complex number z = a + bi can be visually represented as a pair of numbers (a, b) forming a position vector (blue) or a point (red) on an Argand diagram, representing the complex plane.
Matrix (mathematics)Array of numbersAn m × n matrix: the m rows are horizontal and the n columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts.Analytic function of a matrixIn mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a matrix, which is involved in the closed-form solution of systems of linear differential equations.
Power seriesIn mathematics, a power series (in one variable) is an infinite series of the form ∑n=0∞an(x−c)n=a0+a1(x−c)+a2(x−c)2+…{\displaystyle \sum _{n=0}^{\infty }a_{n}\left(x-c ight)^{n}=a_{0}+a_{1}(x-c)+a_{2}(x-c)^{2}+\dots } where an{\displaystyle a_{n}} represents the coefficient of the nth term and c is a constant called the center of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions.
Real numberIn mathematics, a real number is a number that can be used to measure a continuous one-dimensionalquantity such as a length, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.
Lie algebraAlgebraic structure used in analysisLie groupGroup that is also a differentiable manifold with group operations that are smooth
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.*As an Amazon Associate I earn from qualifying purchases.