Eigenvalues and eigenvectorsIn linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.
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.
Generalized eigenvectorIn linear algebra, a generalized eigenvector of an matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let be an -dimensional vector space and let be the matrix representation of a linear map from to with respect to some ordered basis. There may not always exist a full set of linearly independent eigenvectors of that form a complete basis for . That is, the matrix may not be diagonalizable.
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 .
Defective matrixIn linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, an n × n matrix is defective if and only if it does not have n linearly independent eigenvectors. A complete basis is formed by augmenting the eigenvectors with generalized eigenvectors, which are necessary for solving defective systems of ordinary differential equations and other problems.
Diagonalizable matrixIn linear algebra, a square matrix is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix and a diagonal matrix such that , or equivalently . (Such , are not unique.) For a finite-dimensional vector space , a linear map is called diagonalizable if there exists an ordered basis of consisting of eigenvectors of .
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.
Transformation matrixIn linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then for some matrix , called the transformation matrix of . Note that has rows and columns, whereas the transformation is from to . There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors. Matrices allow arbitrary linear transformations to be displayed in a consistent format, suitable for computation.
Affine transformationIn Euclidean geometry, an affine transformation or affinity (from the Latin, affinis, "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments.
RandomnessIn common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if the probability distribution is known, the frequency of different outcomes over repeated events (or "trials") is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4.
Ordinary differential equationIn mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations which may be with respect to one independent variable. A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a_0(x), .
Differential equationIn mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Hardware random number generatorIn computing, a hardware random number generator (HRNG), true random number generator (TRNG) or non-deterministic random bit generator (NRBG) is a device that generates random numbers from a physical process capable of producing entropy (in other words, the device always has access to a physical entropy source), rather than by means of an algorithm. Such devices are often based on microscopic phenomena that generate low-level, statistically random "noise" signals, such as thermal noise, the photoelectric effect, involving a beam splitter, and other quantum phenomena.
Eigenvalue algorithmIn numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors. Eigenvalues and eigenvectors and Generalized eigenvector Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.
Linear differential equationIn mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable x. Such an equation is an ordinary differential equation (ODE).
RandomizationRandomization is the process of making something random. Randomization is not haphazard; instead, a random process is a sequence of random variables describing a process whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. For example, a random sample of individuals from a population refers to a sample where every individual has a known probability of being sampled. This would be contrasted with nonprobability sampling where arbitrary individuals are selected.
Hermitian matrixIn mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: or in matrix form: Hermitian matrices can be understood as the complex extension of real symmetric matrices.
Spectral theoremIn mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces.
Homogeneous differential equationA differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written where f and g are homogeneous functions of the same degree of x and y. In this case, the change of variable y = ux leads to an equation of the form which is easy to solve by integration of the two members. Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives.
Orthogonal diagonalizationIn linear algebra, an orthogonal diagonalization of a symmetric matrix is a diagonalization by means of an orthogonal change of coordinates. The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form q(x) on Rn by means of an orthogonal change of coordinates X = PY. Step 1: find the symmetric matrix A which represents q and find its characteristic polynomial Step 2: find the eigenvalues of A which are the roots of . Step 3: for each eigenvalue of A from step 2, find an orthogonal basis of its eigenspace.
