Identity matrixIn linear algebra, the identity matrix of size n{\displaystyle n} is the n×n{\displaystyle n\times n}square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties; for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1.
Simple ringIn mathematics, a simple ring is a non-zero ring that has no two-sided ideals besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a simple ring is necessarily a field. It follows that a simple ring is an associative algebra over this field. It is then called a simple algebra over this field.
Matrix (mathematics)Array of numbersAn m × n matrix: the m rows are horizontal and the n columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts.
Commutative ringIn mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.
Matrix multiplicationIn mathematics, specifically 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 additionIn mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. For a vector, v→{\displaystyle {\vec {v}}\!}, adding two matrices would have the geometric effect of applying each matrix transformation separately onto v→{\displaystyle {\vec {v}}\!}, then adding the transformed vectors.Associative algebraIn mathematics, an associative algebraA over a commutative ring (often a field) K is a ringA together with a ring homomorphism from K into the center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of K).
Rng (algebra)In abstract algebra, a rng (pronounced "rung" ) or non-unital ring or pseudo-ring is an algebraic structure satisfying the same properties as a ring, but without assuming the existence of a multiplicative identity. The term rng is meant to suggest that it is a ring without i, that is, without the requirement for an identity element.
Hilbert spaceType of vector space in mathHierarchy of mathematical spaces. Hilbert spaces are a subset of Inner product spaces, which are a subset of normed vector spaces, which are a subset of metric spaces, which in turn are a subset of topological spaces.The state of a vibrating string can be modeled as a point in a Hilbert space.Ring (mathematics)Algebraic structure with addition and multiplicationIn mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted like addition and multiplication of integers. They work similarly to integer addition and multiplication, except that multiplication in a ring does not need to be commutative.
Abstract algebraIn mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field.Triangular matrixIn mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero. Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis.*As an Amazon Associate I earn from qualifying purchases.