Search results
Results From The WOW.Com Content Network
In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U −1 equals its conjugate transpose U *, that is, if = =, where I is the identity matrix.. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger ( † ), so the equation above is written
In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication.The unitary group is a subgroup of the general linear group GL(n, C), and it has as a subgroup the special unitary group, consisting of those unitary matrices with determinant 1.
In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is matrix multiplication.
More generally, unitary matrices are precisely the unitary operators on finite-dimensional Hilbert spaces, so the notion of a unitary operator is a generalization of the notion of a unitary matrix. Orthogonal matrices are the special case of unitary matrices in which all entries are real. [4] They are the unitary operators on R n.
Specifically, the singular value decomposition of an complex matrix is a factorization of the form =, where is an complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, is an complex unitary matrix, and is the conjugate transpose of . Such decomposition ...
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G.The general theory is well-developed in the case that G is a locally compact topological group and the representations are strongly continuous.
For example, a 2,1 represents the element at the second row and first column of the matrix. In mathematics, a matrix (pl.: matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object.
Several important classes of matrices are subsets of each other. This article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries. Matrices have a long history of both study and application, leading to ...