Matrix (mathematics)In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.
Matrix multiplicationIn mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB.
Matrix exponentialIn 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 X be an n×n real or complex matrix. The exponential of X, denoted by eX or exp(X), is the n×n matrix given by the power series where is defined to be the identity matrix with the same dimensions as .
Logarithm of a matrixIn mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in an element of a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.
Local foodLocal food is food that is produced within a short distance of where it is consumed, often accompanied by a social structure and supply chain different from the large-scale supermarket system. Local food (or locavore) movements aim to connect food producers and consumers in the same geographic region, to develop more self-reliant and resilient food networks; improve local economies; or to affect the health, environment, community, or society of a particular place.
Diagonal matrixIn linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is , while an example of a 3×3 diagonal matrix is. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix. A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size).
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 .
Olympic GamesThe modern Olympic Games or Olympics (Jeux olympiques) are the leading international sporting events featuring summer and winter sports competitions in which thousands of athletes from around the world participate in a variety of competitions. The Olympic Games are considered the world's foremost sports competition with more than 200 teams, representing sovereign states and territories, participating. The Olympic Games are normally held every four years, and since 1994, have alternated between the Summer and Winter Olympics every two years during the four-year period.
Summer Olympic GamesThe Summer Olympic Games (Jeux olympiques d'été), also known as the Games of the Olympiad, and often referred to as the Summer Olympics, is a major international multi-sport event normally held once every four years. The inaugural Games took place in 1896 in Athens, Greece, and the most recent Games were held in 2021 in Tokyo, Japan. The International Olympic Committee (IOC) is responsible for organising the Games and for overseeing the host city's preparations.
Winter Olympic GamesThe Winter Olympic Games (Jeux olympiques d'hiver) is a major international multi-sport event held once every four years for sports practiced on snow and ice. The first Winter Olympic Games, the 1924 Winter Olympics, were held in Chamonix, France. The modern Olympic Games were inspired by the ancient Olympic Games, which were held in Olympia, Greece, from 776 BC to 394 AD. The Baron Pierre de Coubertin of France founded the International Olympic Committee (IOC) 1,500 years later in 1894, leading to the first modern Summer Olympic Games in Athens, Greece in 1896.
