Companion matrixIn linear algebra, the Frobenius companion matrix of the monic polynomial is the square matrix defined as Some authors use the transpose of this matrix, , which is more convenient for some purposes such as linear recurrence relations (see below). is defined from the coefficients of , while the characteristic polynomial as well as the minimal polynomial of are equal to . In this sense, the matrix and the polynomial are "companions". Any matrix A with entries in a field F has characteristic polynomial , which in turn has companion matrix .
Matrix differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to their derivatives. For example, a first-order matrix ordinary differential equation is where is an vector of functions of an underlying variable , is the vector of first derivatives of these functions, and is an matrix of coefficients.
Mixed boundary conditionIn mathematics, a mixed boundary condition for a partial differential equation defines a boundary value problem in which the solution of the given equation is required to satisfy different boundary conditions on disjoint parts of the boundary of the domain where the condition is stated. Precisely, in a mixed boundary value problem, the solution is required to satisfy a Dirichlet or a Neumann boundary condition in a mutually exclusive way on disjoint parts of the boundary.
Eigendecomposition of a matrixIn linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Eigenvalue, eigenvector and eigenspace A (nonzero) vector v of dimension N is an eigenvector of a square N × N matrix A if it satisfies a linear equation of the form for some scalar λ.
Divide-and-conquer eigenvalue algorithmDivide-and-conquer eigenvalue algorithms are a class of eigenvalue algorithms for Hermitian or real symmetric matrices that have recently (circa 1990s) become competitive in terms of stability and efficiency with more traditional algorithms such as the QR algorithm. The basic concept behind these algorithms is the divide-and-conquer approach from computer science. An eigenvalue problem is divided into two problems of roughly half the size, each of these are solved recursively, and the eigenvalues of the original problem are computed from the results of these smaller problems.
Jacobi elliptic functionsIn mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for .
Theta functionIn mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory. The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function.
Transcendental functionIn mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed algebraically. Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions. Formally, an analytic function f (z) of one real or complex variable z is transcendental if it is algebraically independent of that variable.
Linear mapIn mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a .
Scalar multiplicationIn mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector—without changing its direction. The term "scalar" itself derives from this usage: a scalar is that which scales vectors.
Linear algebraLinear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
Variation of parametersIn mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that involve guessing and do not work for all inhomogeneous linear differential equations.
