Row equivalenceIn linear algebra, two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Alternatively, two m × n matrices are row equivalent if and only if they have the same row space. The concept is most commonly applied to matrices that represent systems of linear equations, in which case two matrices of the same size are row equivalent if and only if the corresponding homogeneous systems have the same set of solutions, or equivalently the matrices have the same null space.
Gamma matricesIn mathematical physics, the gamma matrices, also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts.
Leibniz formula for determinantsIn algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements. If is an matrix, where is the entry in the -th row and -th column of , the formula is where is the sign function of permutations in the permutation group , which returns and for even and odd permutations, respectively. Another common notation used for the formula is in terms of the Levi-Civita symbol and makes use of the Einstein summation notation, where it becomes which may be more familiar to physicists.
Minor (linear algebra)In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which in turn are useful for computing both the determinant and inverse of square matrices. The requirement that the square matrix be smaller than the original matrix is often omitted in the definition.
Higher-dimensional gamma matricesIn mathematical physics, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional Gamma matrices of Dirac, which are a mainstay of relativistic quantum mechanics. They are utilized in relativistically invariant wave equations for fermions (such as spinors) in arbitrary space-time dimensions, notably in string theory and supergravity. The Weyl–Brauer matrices provide an explicit construction of higher-dimensional gamma matrices for Weyl spinors.
Elementary symmetric polynomialIn mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial P is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials.
IPadThe iPad is a brand of iOS and iPadOS-based tablet computers that are developed by Apple Inc, first introduced on January 27, 2010. The iPad range consists of the original iPad lineup and the flagship products iPad Mini, iPad Air, and iPad Pro. The iPhone's iOS operating system (OS) was initially used for the iPad but in September 2019, its OS was switched to a fork of iOS called iPadOS that has better support for the device's hardware and its user interface is customized for the tablets' larger screens.
IPad ProThe iPad Pro is a premium model of Apple's iPad tablet computer. It runs iPadOS, a tablet-optimized version of the iOS operating system. The original iPad Pro was introduced in September 2015, and ran iOS 9. It had an A9X chip, and came in two sizes: 9.7-inch and 12.9 inch. The second-generation iPad Pro, unveiled in June 2017, had an upgraded A10X Fusion chip and swapped the 9.7-inch screen for a larger 10.5-inch display. The third-generation iPad Pro, announced in October 2018, eliminated the home button, and featured Face ID; it came in 11-inch and 12.
IPad MiniThe iPad Mini (branded and marketed as iPad mini) is a line of mini tablet computers designed, developed, and marketed by Apple Inc. It is a sub-series of the iPad line of tablets, with screen sizes of 7.9 inches and 8.3 inches. The first-generation iPad Mini was announced on October 23, 2012, and was released on November 2, 2012, in nearly all of Apple's markets. It featured similar internal specifications to the iPad 2, including its display resolution.
