In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element a has a multiplicative inverse, that is, an element usually denoted a–1, such that a a–1 = a–1 a = 1. So, (right) division may be defined as a / b = a b–1, but this notation is avoided, as one may have a b–1 ≠ b–1 a.
Linear algebra is the branch of mathematics concerning linear equations such as:
In linear algebra, the trace of a square matrix A, denoted tr(A), is defined to be the sum of elements on the main diagonal of A. The trace is only defined for a square matrix.
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order . Any two square matrices of the same order can be added and multiplied.
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of orthogonal matrices, where the group operation is given by matrix multiplication. The orthogonal group is an algebraic group and a Lie group. It is compact.
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. 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, is a diagonal matrix.
In mathematics, an algebra over a field is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.
In linear algebra, two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix P such that
In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix is skew-Hermitian if it satisfies the relation
In linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above the first superdiagonal. They are named after Karl Hessenberg.
In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal/lower diagonal, and the supradiagonal/upper diagonal. For example, the following matrix is tridiagonal:
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if is the symmetric matrix that defines the quadratic form, and is any invertible matrix such that is diagonal, then the number of negative elements in the diagonal of is always the same, for all such ; and the same goes for the number of positive elements.
In abstract algebra, the biquaternions are the numbers w + xi + yj + zk, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof:
In linear algebra, an eigenvector 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.
In mathematics, a matrix 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.
In linear algebra, two matrices and are said to commute if , or equivalently if their commutator is zero. A set of matrices is said to commute if they commute pairwise, meaning that every pair of matrices in the set commute with each other.
In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined on matrices, functions of matrices, and the eigenvalues of matrices.
