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.
Permutation matrixIn mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say P, represents a permutation of m elements and, when used to multiply another matrix, say A, results in permuting the rows (when pre-multiplying, to form PA) or columns (when post-multiplying, to form AP) of the matrix A.
Markov chainA Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs now." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC).
Stationary distributionStationary distribution may refer to: A special distribution for a Markov chain such that if the chain starts with its stationary distribution, the marginal distribution of all states at any time will always be the stationary distribution. Assuming irreducibility, the stationary distribution is always unique if it exists, and its existence can be implied by positive recurrence of all states. The stationary distribution has the interpretation of the limiting distribution when the chain is irreducible and aperiodic.
Material conditionalThe material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol is interpreted as material implication, a formula is true unless is true and is false. Material implication can also be characterized inferentially by modus ponens, modus tollens, conditional proof, and classical reductio ad absurdum. Material implication is used in all the basic systems of classical logic as well as some nonclassical logics.
Euclidean vectorIn mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a directed line segment, or graphically as an arrow connecting an initial point A with a terminal point B, and denoted by . A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier".
Constant-recursive sequenceIn mathematics and theoretical computer science, a constant-recursive sequence is an infinite sequence of numbers where each number in the sequence is equal to a fixed linear combination of one or more of its immediate predecessors. A constant-recursive sequence is also known as a linear recurrence sequence, linear-recursive sequence, linear-recurrent sequence, a C-finite sequence, or a solution to a linear recurrence with constant coefficients.
Recurrence relationIn mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation. If the values of the first numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In linear recurrences, the nth term is equated to a linear function of the previous terms.
Coordinate vectorIn linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensional Cartesian coordinate system with the basis as the axes of this system. Coordinates are always specified relative to an ordered basis. Bases and their associated coordinate representations let one realize vector spaces and linear transformations concretely as column vectors, row vectors, and matrices; hence, they are useful in calculations.
Camel caseCamel case (sometimes stylized as camelCase or CamelCase, also known as camel caps or more formally as medial capitals) is the practice of writing phrases without spaces or punctuation and with capitalized words. The format indicates the first word starting with either case, then the following words having an initial uppercase letter. Common examples include "YouTube", "iPhone" and "eBay". Camel case is often used as a naming convention in computer programming.
Letter caseLetter case is the distinction between the letters that are in larger uppercase or capitals (or more formally majuscule) and smaller lowercase (or more formally minuscule) in the written representation of certain languages. The writing systems that distinguish between the upper- and lowercase have two parallel sets of letters: each in the majuscule set has a counterpart in the minuscule set. Some counterpart letters have the same shape, and differ only in size (e.g. {C,c} or {S,s}), but for others the shapes are different (e.
Dimension (vector space)In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say is if the dimension of is finite, and if its dimension is infinite.
