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In linear algebra, a matrix unit is a matrix with only one nonzero entry with value 1. [1] [2] The matrix unit with a 1 in the ith row and jth column is denoted as .For example, the 3 by 3 matrix unit with i = 1 and j = 2 is = [] A vector unit is a standard unit vector.
U can be written as U = e iH, where e indicates the matrix exponential, i is the imaginary unit, and H is a Hermitian matrix. For any nonnegative integer n, the set of all n × n unitary matrices with matrix multiplication forms a group, called the unitary group U(n). Every square matrix with unit Euclidean norm is the average of two unitary ...
The matrix A is said to represent the linear map f, and A is called the transformation matrix of f. For example, the 2×2 matrix = [] can be viewed as the transform of the unit square into a parallelogram with vertices at (0, 0), (a, b), (a + c, b + d), and (c, d).
In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. When A is an invertible matrix there is a matrix A −1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using ...
For example, to produce one unit of intermediate good , one unit of basic commodity , two units of , no units of , and one unit of are needed, corresponding to the first column of . Using matrix multiplication, compute
An "almost" triangular matrix, for example, an upper Hessenberg matrix has zero entries below the first subdiagonal. Hollow matrix: A square matrix whose main diagonal comprises only zero elements. Integer matrix: A matrix whose entries are all integers. Logical matrix: A matrix with all entries either 0 or 1.
The term unit matrix is ambiguous, because it is also used for a matrix of ones and for any unit of the ring of all matrices. [ 7 ] In some fields, such as group theory or quantum mechanics , the identity matrix is sometimes denoted by a boldface one, 1 {\displaystyle \mathbf {1} } , or called "id" (short for identity).
The all-ones matrix arises in the mathematical field of combinatorics, particularly involving the application of algebraic methods to graph theory.For example, if A is the adjacency matrix of an n-vertex undirected graph G, and J is the all-ones matrix of the same dimension, then G is a regular graph if and only if AJ = JA. [7]