|
Unitary Invariance
Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigroup Politics * Unitary authority * Unitary state See also * Unital (other) * Unitarianism, belief that God is one entity * * {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Divisor
In mathematics, a natural number ''a'' is a unitary divisor (or Hall divisor) of a number ''b'' if ''a'' is a divisor of ''b'' and if ''a'' and \frac are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and \frac=12 have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and \frac=10 have a common factor other than 1, namely 2. 1 is a unitary divisor of every natural number. Equivalently, a divisor ''a'' of ''b'' is a unitary divisor if and only if every prime number, prime factor of ''a'' has the same multiplicity (mathematics), multiplicity in ''a'' as it has in ''b''. The sum-of-unitary-divisors function is denoted by the lowercase Greek letter sigma thus: σ*(''n''). The sum of the ''k''-th exponentiation, powers of the unitary divisors is denoted by σ*''k''(''n''): :\sigma_k^*(n) = \sum_ \!\! d^k. If the proper divisor, proper unitary divisors of a given number add up to that number, then that num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Element
In mathematics, an element ''x'' of a *-algebra is unitary if it satisfies x^* = x^. In functional analysis, a linear operator ''A'' from a Hilbert space into itself is called unitary if it is invertible and its inverse is equal to its own adjoint ''A'' and that the domain of ''A'' is the same as that of ''A''. See unitary operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator ''A'' is unitary if and only if the matrix describing ''A'' with respect to this basis is a unitary matrix In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is .... See also * * * References * * * {{DEFAULTSORT:Self-Adjoint Abstract algebra Linear algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Group
In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group. In the simple case , the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1, under multiplication. All the unitary groups contain copies of this group. The unitary group U(''n'') is a real Lie group of dimension ''n''2. The Lie algebra of U(''n'') consists of skew-Hermitian matrices, with the Lie bracket given by the commutator. The general unitary group (also called the group of unitary similitudes) consists of all matrices ''A'' such that ''A''∗''A'' is a nonzero multiple of the identity matrix, and is just the product of the unitary gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Matrix
In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where 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 U^\dagger U = UU^\dagger = I. The real analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes. Properties For any unitary matrix of finite size, the following hold: * Given two complex vectors and , multiplication by preserves their inner product; that is, . * is normal (U^* U = UU^*). * is diagonalizable; that is, is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus, has a decomposition of the form U = VDV^*, where is unitary, and is diagonal and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Morphism
In category theory, a branch of mathematics, a dagger category (also called involutive category or category with involution) is a category equipped with a certain structure called ''dagger'' or ''involution''. The name dagger category was coined by Peter Selinger. Formal definition A dagger category is a category \mathcal equipped with an involutive contravariant endofunctor \dagger which is the identity on objects. In detail, this means that: * for all morphisms f: A \to B, there exist its adjoint f^\dagger: B \to A * for all morphisms f, (f^\dagger)^\dagger = f * for all objects A, \mathrm_A^\dagger = \mathrm_A * for all f: A \to B and g: B \to C, (g \circ f)^\dagger = f^\dagger \circ g^\dagger: C \to A Note that in the previous definition, the term "adjoint" is used in a way analogous to (and inspired by) the linear-algebraic sense, not in the category-theoretic sense. Some sources define a category with involution to be a dagger category with the additional property t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the concept of isomorphism ''between'' Hilbert spaces. A unitary element is a generalization of a unitary operator. In a unital algebra, an element of the algebra is called a unitary element if , where is the identity element. Definition Definition 1. A ''unitary operator'' is a bounded linear operator on a Hilbert space that satisfies , where is the adjoint of , and is the identity operator. The weaker condition defines an ''isometry''. The other condition, , defines a ''coisometry''. Thus a unitary operator is a bounded linear operator which is both an isometry and a coisometry, or, equivalently, a surjective isometry. An equivalent definition is the following: Definition 2. A ''unitary operator'' is a bounded linear operator on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Transformation
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation. Formal definition More precisely, a unitary transformation is an isomorphism between two inner product spaces (such as Hilbert spaces). In other words, a ''unitary transformation'' is a bijective function U : H \to H_2\, between two inner product spaces, H and H_2, such that \langle Ux, Uy \rangle_ = \langle x, y \rangle_ \quad \text x, y \in H. Properties A unitary transformation is an isometry, as one can see by setting x=y in this formula. Unitary operator In the case when H_1 and H_2 are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator. Antiunitary transformation A closely related notion is that of antiunitary transformation, which is a bijective function :U:H_1\to H_2\, between two co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Representation
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 case ''G'' is a locally compact ( Hausdorff) topological group and the representations are strongly continuous. The theory has been widely applied in quantum mechanics since the 1920s, particularly influenced by Hermann Weyl's 1928 book ''Gruppentheorie und Quantenmechanik''. One of the pioneers in constructing a general theory of unitary representations, for any group ''G'' rather than just for particular groups useful in applications, was George Mackey. Context in harmonic analysis The theory of unitary representations of topological groups is closely connected with harmonic analysis. In the case of an abelian group ''G'', a fairly complete picture of the representation theory of ''G'' is given by Pontryagin duality. In general, the unitary equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitarity (physics)
In quantum physics, unitarity is the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. This is typically taken as an axiom or basic postulate of quantum mechanics, while generalizations of or departures from unitarity are part of speculations about theories that may go beyond quantum mechanics. A unitarity bound is any inequality that follows from the unitarity of the evolution operator, i.e. from the statement that time evolution preserves inner products in Hilbert space. Hamiltonian evolution Time evolution described by a time-independent Hamiltonian is represented by a one-parameter family of unitary operators, for which the Hamiltonian is a generator: U(t) = e^. In the Schrödinger picture, the unitary operators are taken to act upon the system's quantum state, whereas in the Heisenberg picture, the time dependence is incorporated into the observables instead. Implications of unit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E-unitary Inverse Semigroup
In group theory, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a semigroup in which every element ''x'' in ''S'' has a unique ''inverse'' ''y'' in ''S'' in the sense that ''x = xyx'' and ''y = yxy'', i.e. a regular semigroup in which every element has a unique inverse. Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries. (The convention followed in this article will be that of writing a function on the right of its argument, e.g. ''x f'' rather than ''f(x)'', and composing functions from left to right—a convention often observed in semigroup theory.) Origins Inverse semigroups were introduced independently by Viktor Vladimirovich Wagner in the Soviet Union in 1952, and by Gordon Preston in the United Kingdom in 1954. Both authors arrived at inverse semigroups via the study of partial bijections of a set: a partial transformation ''α'' of a set ''X'' is a function from ''A'' to '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Authority
A unitary authority is a local authority responsible for all local government functions within its area or performing additional functions that elsewhere are usually performed by a higher level of sub-national government or the national government. Typically unitary authorities cover towns or cities which are large enough to function independently of a council or other authority. An authority can be a unit of a county or combined authority. Canada In Canada, each province creates its own system of local government, so terminology varies substantially. In certain provinces (e.g. Alberta, Nova Scotia) there is ''only'' one level of local government in that province, so no special term is used to describe the situation. British Columbia has only one such municipality, Northern Rockies Regional Municipality, which was established in 2009. In Ontario the term single-tier municipalities is used, for a similar concept. Their character varies, and while most function as cities with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary State
A unitary state is a sovereign state governed as a single entity in which the central government is the supreme authority. The central government may create (or abolish) administrative divisions (sub-national units). Such units exercise only the powers that the central government chooses to delegate. Although political power may be delegated through devolution to regional or local governments by statute, the central government may abrogate the acts of devolved governments or curtail (or expand) their powers. Unitary states stand in contrast with federations, also known as ''federal states''. A large majority of the world's sovereign states (166 of the 193 UN member states) have a unitary system of government. Devolution compared with federalism A unitary system of government can be considered the opposite of federalism. In federations, the provincial/regional governments share powers with the central government as equal actors through a written constitution, to which the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
