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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 ...
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.
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]
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
An Toeplitz matrix may be defined as a matrix where , =, for constants , …,. The set of n × n {\displaystyle n\times n} Toeplitz matrices is a subspace of the vector space of n × n {\displaystyle n\times n} matrices (under matrix addition and scalar multiplication).
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.
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In algebra, a unit or invertible element [a] of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that v u = u v = 1 , {\displaystyle vu=uv=1,} where 1 is the multiplicative identity ; the element v is unique for this property and is called the ...