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entries are said to be unitarily equivalent if there exists a unitary matrix U such that B = U *AU. Two matrices which are unitarily equivalent are also...

normal if A*A = AA*. One can show that A is normal if and only if it is unitarily diagonalizable using the Schur decomposition. That is, any matrix can...

Unitary equivalence may refer to: Unitary equivalence of bounded operators in Hilbert space; see self-adjoint operator Unitary equivalence of a unitary...

decomposition. It allows one to write an arbitrary complex square matrix as unitarily similar to an upper triangular matrix whose diagonal elements are the...

Unitarian Communists (Movimento dei Comunisti Unitari, MCU), or simply Unitarian Communists (Italian: Comunisti Unitari), was a communist political party in Italy...

case. That is to say, operators are self-adjoint if and only if they are unitarily equivalent to real-valued multiplication operators. With suitable modifications...

unitary matrix then A and B are unitarily equivalent. The spectral theorem says that every normal matrix is unitarily equivalent to some diagonal matrix...

According to quantum mechanics, a quantum system can only either evolve unitarily according to the Schrödinger equation, or be measured (sometimes called...

In the many-worlds interpretation, the universal wave function evolves unitarily without collapse. Interactions lead to decoherence, producing dynamically...

Assembly of Haiti.[clarification needed] The government is organized unitarily, thus the central government delegates powers to the departments without...

representations are unitarily inequivalent; and any irreducible representation which is not trivial on the center of Hn is unitarily equivalent to exactly...

norm is reserved for sub-multiplicative norms. A matrix norm is called unitarily invariant if for all unitary matrices U , V {\displaystyle U,V} and matrix...

of the Holy Scripture could enlight its spiritual meaning to have been unitarily inspired by God. Augustine's theological views extended into the centuries...

by a trace-class operator, then their absolutely continuous parts are unitarily equivalent. In particular if a self-adjoint operator has absolutely continuous...

Stone–von Neumann theorem, the unitary representations ρ and ρ ∘ j are unitarily equivalent, so there is a unique intertwiner W ∈ U(L2(R)) such that ρ...

pre-norm is a norm in the first major paper discussing the theory of unitarily invariant norms and symmetric gauge functions (now known as symmetric...

equipped with the Fell topology. Informally, this means that if G acts unitarily on a Hilbert space and has "almost invariant vectors", then it has a nonzero...

Hilbert space. The Wightman axioms require that the Poincaré group acts unitarily on the Hilbert space. In other words, a change of reference frame (position...

the same, but different field theories correspond to different (i.e., unitarily inequivalent) representations. Philosophically, the action of the CCR...

identified with a wavefunction, and this wavefunction always evolves unitarily, i.e., following the Schrödinger equation with no collapses. Consequently...

spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix; implying in particular that any matrix A...

C. Eckart and G. Young. L. Mirsky generalized the result to arbitrary unitarily invariant norms. Let D = U Σ V ⊤ ∈ R m × n , m ≥ n {\displaystyle D=U\Sigma...

∗ {\displaystyle U^{*}U=UU^{*}} ). U is diagonalizable; that is, U is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem...

is a proper subset of matrices that are unitarily diagonalizable. A matrix M {\displaystyle M} is unitarily diagonalizable if and only if it is normal...

finite field are unitarily congruent to the standard one, represented by the identity matrix; that is, any Hermitian form is unitarily equivalent to Ψ...

are "navigable waters" under the Clean Water Act should be considered "unitarily" for purposes of NPDES permitting. The Miccosukee Tribe challenged the...

commit. If the protocol is unconditionally concealing, then Alice can unitarily transform these states into each other using the properties of the Schmidt...

consequence of the spectral theorem for normal operators. By the theorem, U is unitarily equivalent to multiplication by a Borel-measurable f on L2(μ), for some...

the absence of outside forces or interactions, a quantum state evolves unitarily over time. Consequently, a pure quantum state remains pure. However, if...

other. In technical terms, this comes about because the two vacua lead to unitarily inequivalent representations of the quantum field canonical commutation...

Neumann theorem, the Heisenberg picture and the Schrödinger picture are unitarily equivalent, just a basis change in Hilbert space. In some sense, the Heisenberg...

unitary. Physical symmetries act on the Hilbert space of quantum states unitarily or antiunitarily due to Wigner's theorem (supersymmetry is another matter...

projection-valued measures on (X, M) with values in the projections of H, K. π, ρ are unitarily equivalent if and only if there is a unitary operator U:H → K such that...

theorem that states that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication operator on an L2 space. These operators...

matrix, and thus also square, the spectral theorem ensures that it can be unitarily diagonalized using a basis of eigenvectors, and thus decomposed as ⁠ M...

convenience one obtains the 'space-translated' Hamiltonian, which is unitarily equivalent to the lab-frame Hamiltonian, which contains the original potential...

p=\infty } is then q = 1 {\displaystyle q=1} . The Schatten norms are unitarily invariant: for unitary operators U {\displaystyle U} and V {\displaystyle...

spaces H and G respectively, are each unitarily equivalent to a subrepresentation of the other, then they are unitarily equivalent. A proof that resembles...

One can show that A {\displaystyle A} is normal if and only if it is unitarily diagonalizable: By the Schur decomposition, we have A = U T U ∗ {\displaystyle...

a\in A} . Then π {\displaystyle \pi } , π ′ {\displaystyle \pi '} are unitarily equivalent ∗ {\displaystyle *} -representations i.e. there is a unitary...

Two unitary representations π1: G → U(H1), π2: G → U(H2) are said to be unitarily equivalent if there is a unitary transformation A:H1 → H2 such that π1(g)...

(up to a finite precision) but the problem manifests if the qubit is unitarily transformed (e.g. by the Hadamard quantum gate) to be polarised (which...

disjoint when the operator T is of uniform multiplicity, say m, i.e. if T is unitarily equivalent to multiplication by λ on the direct sum ⨁ i = 1 m L 2 ( R...

{(f_{1},f_{2})=\sum _{g,h}(\Phi (h^{-1}g)f_{1}(g),f_{2}(h)).}} G acts unitarily on H {\displaystyle {\mathcal {H}}} by U ( g ) f ( x ) = f ( g − 1 x )...

of pairs of matrices. Moreover, any square matrix with zero trace is unitarily equivalent to a square matrix with diagonal consisting of all zeros. The...

\mathbb {C} ^{n}.} A square matrix is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues. Hermitian matrices are fundamental...

state, the absence of a natural notion of particle and the appearance of unitarily inequivalent representations of the algebra of observables. Using perturbation...

Neumann theorem, the Heisenberg picture and the Schrödinger picture must be unitarily equivalent, as detailed below. Matrix mechanics rapidly developed into...

particular, a normal operator on a finite-dimensional inner product space) is unitarily diagonalizable. Let T {\displaystyle T} be a bounded operator. The following...

non-unitary. Penrose claims that quantum systems will in fact no longer evolve unitarily as soon as gravitation comes into play, precisely as in black holes. The...

generalizations of the arithmetic geometric mean inequality apply on the level of unitarily invariant norms, since, even if the matrices A {\displaystyle A} and B...

more is true. The spectral theorem says that every normal matrix is unitarily similar to a diagonal matrix (if AA∗ = A∗A then there exists a unitary...

\Psi _{\alpha }^{-})=\delta (\beta -\alpha ),} and the three sets are unitarily equivalent. Now rewrite the eigenvalue equation, ( E α − H 0 ± i ϵ ) Ψ...

satisfying the canonical commutation relation, and shows that these are all unitarily equivalent to the position operator and momentum operator on L 2 ( R )...

there is a standard Borel space X and a measure μ on X such that it is unitarily equivalent as an operator algebra to L∞μ(X) acting on a direct integral...

Vandermonde determinant. If f : V n → C {\textstyle f:V_{n}\to \mathbb {C} } is unitarily invariant, with sufficient regularity and decay, then it can be decomposed...

localized in standard QFT. A third, less obvious, complication is that "unitarily inequivalent representations" are endemic in QFT; for example, the same...

that any square matrix A has a Schur decomposition. This means that A is unitarily equivalent (i.e. similar, using a unitary matrix as change of basis) to...

and B are Hermitian matrices and ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is a unitarily invariant norm, then (Bhatia 1997, Theorem IX.3.7) ‖ e A + B ‖ ≤ ‖ e A...

semigroups, Proc. Royal Irish Acad., 97A (1997), 131—137 (with K. Abodayeh). Unitarily-invariant linear spaces in C*-algebras, Proc. Amer. Math. Soc., 126 (1998)...

Proposition—Given any two normalized crinkled arcs in a Hilbert space, each is unitarily equivalent to a reparameterization of the other. If h : [ 0 , 1 ] → [...

representation of the Heisenberg group) is of this form, or more properly unitarily conjugate to the standard one. Further, the group algebra of (the dual...

VV^{*}} has kernel min ( x , y ) {\displaystyle \min(x,y)} . They are unitarily equivalent via U f ( x ) = f ( 1 − x ) {\displaystyle Uf(x)=f(1-x)} ,...

example a quantum quench that is an initially prepared system evolves unitarily in time by a sudden change in the parameters of the initial Hamiltonian...

(in analogy with the fact that irreducible representations are either unitarily equivalent or disjoint). This can be understood through the correspondence...

{R} )} . Here, h is the Planck constant. Each such representation is unitarily inequivalent. The corresponding theta representation is: M ( a , 0 , 0...

Schur's lemma in linear algebra says that every square complex matrix is unitarily triangularizable, see Schur decomposition Schur test for boundedness of...

including the interaction Hamiltonian. Thus, the system will evolve entirely unitarily (without any form of wavefunction collapse) relative to O ′ {\displaystyle...

spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix. Weakening the normality condition to EPness...

singular-value decomposition. If A {\displaystyle A} is normal, then it is unitarily equivalent to a diagonal matrix: A = V Λ V ∗ {\displaystyle A=V\Lambda...

Moscow, Russia Movement of Unitarian Communists (Movimento dei Comunisti Unitari), an Italian communist party Modern Churchpeople's Union, an Anglican liberal...

with respect to τ is identical to the hull-kernel closure of X. Since unitarily equivalent representations have the same kernel, the map π ↦ ker(π) factors...

similar manner the theory of logarithmic norms can also be extended to unitarily invariant norms, e.g. the Hilbert-Schmidt norm for trace class operators...

is that there is a Hilbert space, upon which the Poincaré group acts unitarily. In this way, the concepts of energy, momentum, angular momentum and center...

assemblies, often organised by autonomous rank-and-file committees (Comitati Unitari di Base, CUBs) outside the control of the official unions, became widespread...

group-invariant measure, so it is used to average properties that are not unitarily invariant over all states, or over all unitaries. A particularly widely...

shift operators, implies that the operators L(t) for different t are unitarily equivalent. The matrix L ( t ) {\displaystyle L(t)} has the property that...

}\mathrm {d} ^{3}\mathbf {r} .} The position and momentum operators are unitarily equivalent, with the unitary operator being given explicitly by the Fourier...

1987, Def 17.7). H 2 {\displaystyle H^{2}} is a Hilbert space, and it is unitarily equivalent to ℓ 2 ( N ) {\displaystyle \ell ^{2}(\mathbb {N} )} via the...

the diagonal matrices in SU(1,1). Since evidently K acts unitarily on Hσ and A acts unitarily on H 'σ, both representations are unitary. The representations...

Corollary If two normal operators M and N are similar, then they are unitarily equivalent. Proof: Suppose MS = SN where S is a bounded invertible operator...

Swahn noted that, in Lithuanian and Latvian variants, the formula is "unitarily formed", which is later learned by the snake husband's brothers-in-law...

h(T)e_{k}=h(\lambda _{k})e_{k}.} Generally, any self-adjoint operator T is unitarily equivalent to a multiplication operator; this means that for many purposes...

{\displaystyle U} on L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} is unitarily equivalent to the right translation R {\displaystyle R} on H 1 {\displaystyle...

measure-preserving transformations of (X, μ). The group therefore acts unitarily on the Hilbert space H = L2(X, μ) according to the formula U g f ( x )...

Journal, vol 2, 2002, pp. 769–798, 806. with Ken Dykema, Gary Weiss: Unitarily invariant trace extensions beyond the trace class. In: Complex analysis...

representation theory relies on the following two results: If G = SL(2,R) acts unitarily on a Hilbert space H and ξ is a unit vector fixed by the subgroup N of...

jfa.2010.05.018, S2CID 5570062 Qiu, L.; Zhang, Y.; Li, C.-K. (2005), "Unitarily invariant metrics on the Grassmann space" (PDF), SIAM Journal on Matrix...

court, being that both of which may be aggregated, in that case being unitarily designated "administrative and tax courts". At the present, only the Lisbon...

dell'Ottocento – Percorso 4 Dimore altoborghesi e monumenti a Genova nei decenni unitari Archived 2015-05-18 at the Wayback Machine, from the Genoa Museums website...

criminal proceedings were relevant. However, HMRC could not issue demands unitarily without any time limit as this would be in breach of the fundamental principle...

g_{4}\right]=\left[g_{3},g_{4}\right]=0.} The above set of generators is unitarily equivalent to the following canonical generators: X I I I Z I I I I Z...

{\displaystyle A} . Numerical radius r ( ⋅ ) {\displaystyle r(\cdot )} is a unitarily invariant norm on the space of n × n {\displaystyle n\times n} matrices...

below. Von Neumann algebras A on H, B on K are spatially isomorphic (or unitarily isomorphic) if and only if there is a unitary operator U: H → K such that...

superpositionality to explain how Jewish racial identity, rather than being unitarily white or non-white, is both at the same time. He writes that Jewish racial...

quaternionic matrix A ∈ M n ( H ) {\displaystyle A\in M_{n}(\mathbb {H} )} is unitarily similar to an upper triangular matrix whose diagonal entries correspond...

encoded with this information (i.e. the phase factor) and thus evolve unitarily under this dephasing process, by defining the following encoded qubits:...

shows that they must exist. He also shows that a quaternion matrix is unitarily-equivalent to a triangular matrix. In 1956 he became a Senior Mathematician...

(M_{z}f)(z)=zf(z)} then S {\displaystyle S} and M z {\displaystyle M_{z}} are unitarily equivalent via the unitary map U : ℓ 2 → H 2 {\displaystyle U:\ell ^{2}\to...

principal series are irreducible and two representations πλ and πμ are unitarily equivalent if and only if μ = σ(λ) for some σ in the Weyl group of A....

state) do not span a spin-complete manifold, i.e. it is impossible to unitarily transform the set of all single excitations of an open-shell reference...

on Hilbert spaces H 1 , H 2 {\displaystyle H_{1},H_{2}} are said to be unitarily equivalent if there is a unitary operator U : H 1 → H 2 {\displaystyle...

Neumann theorem, the Heisenberg picture and the Schrödinger picture are unitarily equivalent. In some sense, the Heisenberg picture is more natural and...

over Haar measure). Since the trace (and hence the relative entropy) is unitarily invariant, inequality (3) now follows from (2). This theorem is due to...

the spectral theorem for normal matrices states every normal matrix is unitarily diagonalizable. This leads to a definition of f(T) when T is normal. One...

originated from the Duchy of Modena. These organizations were named the Unitari, Guelfi, and Carbonari. However, Neipperg expressed skepticism about the...

CUB delegati INPS Rete ControG8 Rete Lilliput Rappresentati Sindacali Unitari Sinistra Giovanile Sondagenova Suore del Buon Pastore Time for Peace -...