In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. This is also the signed volume of the n-dimensional parallelepiped spanned by the column or row vectors of the matrix. The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of n-space.
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It is usually understood as a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix. The method is named after Carl Friedrich Gauss (1777–1855), although it was known to Chinese mathematicians as early as 179 A.D..
Linear algebra is the branch of mathematics concerning linear equations such as
The Matrix is a 1999 science fiction action film written and directed by The Wachowskis and starring Keanu Reeves, Laurence Fishburne, Carrie-Anne Moss, Hugo Weaving, and Joe Pantoliano. It depicts a dystopian future in which humanity is unknowingly trapped inside a simulated reality called the Matrix, created by thought-capable machines to control humans while using their bodies as an energy source. Hacker and computer programmer Neo learns this truth and "is drawn into a rebellion against the machines", which involves other people who have been freed from the Matrix.
In linear algebra, a symmetric real matrix is said to be positive definite if the scalar is strictly positive for every non-zero column vector of real numbers. Here denotes the transpose of .. When interpreting as the output of an operator, , that is acting on an input, , the property of positive definiteness implies that the output always has a positive inner product with the input, as often observed in physical processes.
In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context. Less frequently, some mathematics books use U or E to represent the identity matrix, meaning "unit matrix" and the German word "Einheitsmatrix", respectively.
An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors, i.e.
In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring. The matrix product is designed for representing the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. In more detail, if A is an n × m matrix and B is an m × p matrix, their matrix product AB is an n × p matrix, in which the m entries across a row of A are multiplied with the m entries down a column of B and summed to produce an entry of AB. When two linear maps are represented by matrices, then the matrix product represents the composition of the two maps.
In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix to any matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics.
Electrology is the practice of electrical hair removal to permanently remove human hair from the body. Electrolysis is the actual process of removing hair using electricity.
In 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. An example of a 2-by-2 diagonal matrix is ; the following matrix is a 3-by-3 diagonal matrix: . An identity matrix of any size, or any multiple of it, will be a diagonal matrix.
In linear algebra, an n-by-n square matrix A is called invertible if there exists an n-by-n square matrix B such that
In numerical analysis and computer science, a sparse matrix or sparse array is a matrix in which most of the elements are zero. By contrast, if most of the elements are nonzero, then the matrix is considered dense. The number of zero-valued elements divided by the total number of elements is called the sparsity of the matrix. Using those definitions, a matrix will be sparse when its sparsity is greater than 0.5.
The Matrix is a science fiction action media franchise created by The Wachowskis, about a group of heroes who fight a desperate war against machine overlords that have enslaved humanity in an extremely sophisticated virtual reality system. The series is most notable for its use of slow motion, which revolutionized action films to come. The series began with the feature film The Matrix (1999), and continued with two sequels, The Matrix Reloaded and The Matrix Revolutions, all written and directed by The Wachowskis and produced by Joel Silver. The franchise is owned by Warner Bros., which distributed the films along with Village Roadshow Pictures. The latter, along with Silver Pictures are the two production companies that worked on all three films.
A nail is a horn-like envelope covering the tips of the fingers and toes in most primates and a few other mammals. Nails are similar to claws in other animals. Fingernails and toenails are made of a tough protective protein called alpha-keratin. This protein is also found in the hooves and horns of different animals.
The dermis or corium is a layer of skin between the epidermis and subcutaneous tissues, that primarily consists of dense irregular connective tissue and cushions the body from stress and strain. It is divided into two layers, the superficial area adjacent to the epidermis called the papillary region and a deep thicker area known as the reticular dermis. The dermis is tightly connected to the epidermis through a basement membrane. Structural components of the dermis are collagen, elastic fibers, and extrafibrillar matrix. It also contains mechanoreceptors that provide the sense of touch and thermoreceptors that provide the sense of heat. In addition, hair follicles, sweat glands, sebaceous glands, apocrine glands, lymphatic vessels, nerves and blood vessels are present in the dermis. Those blood vessels provide nourishment and waste removal for both dermal and epidermal cells.
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, then v is an eigenvector of T if T(v) is a scalar multiple of v. This condition can be written as the equation
Mechanotransduction is any of various mechanisms by which cells convert mechanical stimulus into electrochemical activity. This form of sensory transduction is responsible for a number of senses and physiological processes in the body, including proprioception, touch, balance, and hearing. The basic mechanism of mechanotransduction involves converting mechanical signals into electrical or chemical signals.
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 × 3, because there are two rows and three columns:
Malignant pilomatricoma is a cutaneous condition characterized by a locally aggressive tumor composed of hair matrix cells.
