Search results
Results From The WOW.Com Content Network
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.
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.
For example, a matrix such that all entries of a row (or a column) are 0 does not have an inverse. If it exists, the inverse of a matrix A is denoted A −1, and, thus verifies = =. A matrix that has an inverse is an invertible matrix.
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q T Q = Q Q T = I , {\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,} where Q T is the transpose of Q and I is the identity matrix .
A matrix whose elements above the main diagonal are all zero is called a lower triangular matrix, while a matrix whose elements below the main diagonal are all zero is called an upper triangular matrix. As with diagonal matrices, the eigenvalues of triangular matrices are the elements of the main diagonal.
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]
In mathematics, a singleton (also known as a unit set [1] or one-point set) is a set with exactly one element. For example, the set { 0 } {\displaystyle \{0\}} is a singleton whose single element is 0 {\displaystyle 0} .