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alternate case: orthonormal basis

Metric signature (1,358 words) [view diff] no match in snippet view article find links to article

In mathematics, the signature of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional

basis) If { e n ( x ) } n {\displaystyle \{e_{n}(x)\}_{n}} is an orthonormal basis for a Hilbert Space with a defined inner product ⟨ f , g ⟩ {\displaystyle

An orthonormal function system (ONS) is an orthonormal basis in a vector space of functions. Melzak, Z. A. (2012), Companion to Concrete Mathematics,

equations are a representation of the Hartree–Fock equation in a non orthonormal basis set which can be of Gaussian-type or Slater-type. It applies to closed-shell

eigenvectors of general (possibly non-Hermitian) matrices by constructing an orthonormal basis of the Krylov subspace, which makes it particularly useful when dealing

{red}{j}} ,\mathbf {\color {green}{k}} )} is a positively oriented orthonormal basis, the basis vectors satisfy the following equalities i × j = k j ×

discovery of Strömberg wavelet and Strömberg's motivation was to find an orthonormal basis for the Hardy spaces. Let m be any non-negative integer. Let V be

Jordan curve. The Gram–Schmidt process can be used to construct an orthonormal basis in the Bergman space and hence an explicit form of the Bergman kernel

four-colored. In 1928, Franklin gave the first description of an orthonormal basis for L²([0,1]) consisting of continuous functions (now known as "Franklin's

(compact) orthogonal group (its isometry group) – and that it admits an orthonormal basis: the change of basis defines the conjugating element conjugating the

n pure second derivatives of f with respect to each vector of an orthonormal basis for Rn. Like the Laplacian, the Laplace–Beltrami operator is defined

to γ {\displaystyle \gamma } are constructed this way. Consider an orthonormal basis of T M {\displaystyle TM} with e 0 {\displaystyle e_{0}} parallel

that it generates n + 1 orthogonal polynomials which can be used as orthonormal basis functions to expand any nth-order polynomial. It is closely related

such systems to string theory. Consider a Hilbert space H with an orthonormal basis of states | p ⟩ {\displaystyle |p\rangle } labelled by the prime numbers

computers using qutrits and qudits in general. A qutrit has three orthonormal basis states or vectors, often denoted | 0 ⟩ {\displaystyle |0\rangle }

and the elements of a spinor space are spinors. After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated

single orthonormal basis of L 2 [ 0 , 1 ] {\displaystyle L^{2}[0,1]} . if { e k } k ∈ Z {\displaystyle \{e_{k}\}_{k\in \mathbb {Z} }} is an orthonormal basis

{\displaystyle {\hat {z}}_{1},\dots ,{\hat {z}}_{\tau }} form an orthonormal basis for F {\displaystyle F} and the vectors y ^ 1 , … , y ^ σ , {\displaystyle

a random function is represented in the eigenbasis, which is an orthonormal basis of the Hilbert space L2 that consists of the eigenfunctions of the

corollary of the Peter–Weyl theorem, is that the characters form an orthonormal basis for the set of square-integrable class functions in K. A second key

U\in B(X,Y)} . Let H {\displaystyle H} be a Hilbert space with an orthonormal basis { e n } {\displaystyle \{e_{n}\}} indexed by the non negative integers

. How was it done? Each pentagram is a 3D quantum system with an orthonormal basis { | A ⟩ , | B ⟩ , | C ⟩ } {\displaystyle \left\{|A\rangle ,|B\rangle

in J where diag(μ(A)) is the diagonalisation of A in an arbitrary orthonormal basis of the separable Hilbert space H. In any two-sided ideal satisfying

wavelet transform, the downsampling is omitted Newland transform, an orthonormal basis of wavelets is formed from appropriately constructed top-hat filters

e_{i}} is an orthonormal basis of span ( L ) {\textstyle {\text{span}}(L)} . (Equivalently, one can declare that, for an orthonormal basis e i {\textstyle

sequence space j within l∞ can be embedded in B(H) using an arbitrary orthonormal basis {en }n=0∞. Associate to a sequence a from j the bounded operator d

{\gamma }}(0)/|{\dot {\gamma }}(0)|} and complete this to get an orthonormal basis { e i ( 0 ) } {\displaystyle {\big \{}e_{i}(0){\big \}}} at T γ (

An electron-on-helium qubit is a quantum bit for which the orthonormal basis states |0⟩ and |1⟩ are defined by quantized motional states or alternatively

space of cusp forms of level Γ 0 {\displaystyle \Gamma _{0}} has an orthonormal basis consisting of simultaneous eigenfunctions for the Hecke operators

e 1 , … , e D − 1 {\displaystyle e_{1},\ldots ,e_{D-1}} forms an orthonormal basis in the simplex. The values x i ∗ , i = 1 , 2 , … , D − 1 {\displaystyle

A}|\langle x,y\rangle |^{2}\leq \|x\|^{2}.} Furthermore, if A is an orthonormal basis for H and x an arbitrary vector, the series ∑ y ∈ A ⟨ x , y ⟩ y {\displaystyle

{\displaystyle \rho } of a group G {\displaystyle G} , and let us work in an orthonormal basis. Thus, ρ {\displaystyle \rho } maps G {\displaystyle G} into the group

operator S can be diagonalized, that it can be represented in some orthonormal basis by a (possibly infinite) matrix of the form [ λ 1 0 ⋯ 0 ⋯ 0 λ 2 ⋯

form in (pseudo-)orthonormal basis: φ ( x , y ) = ± ξ 1 η 1 ± ξ 2 η 2 ± ⋯ ± ξ n η n , ( R ) Bilinear symmetric form in orthonormal basis: φ ( x , y ) =

manifold which is Einstein. Given any point p of M, there exists a gp-orthonormal basis e1, e2, e3, e4 of the tangent space TpM such that the curvature operator

an orthonormal basis of V and let g be an element in GL(V). Applying the Gram–Schmidt process to ge1, ..., gen, there is a unique orthonormal basis f1

defined as follows: let H be a separable complex Hilbert space H with orthonormal basis fn and L(H) the bounded operators on H, consider a linear map α :

of a qubit can be represented by a linear superposition of its two orthonormal basis states (or basis vectors). These vectors are usually denoted as |

_{j=1}^{n}e_{j}{\frac {\partial }{\partial x_{j}}}} where e1, ..., en is an orthonormal basis for Rn, and Rn is considered to be embedded in a complex Clifford

generalization of the conjugate transpose of matrices with respect to an orthonormal basis. Another generalization is available: suppose A {\displaystyle A}

useful for forming direction cosine matrices that express one set of orthonormal basis vectors in terms of another set, or for expressing a known vector

useful for forming direction cosine matrices that express one set of orthonormal basis vectors in terms of another set, or for expressing a known vector

generalization of the conjugate transpose of matrices with respect to an orthonormal basis. Another generalization is available: suppose A {\displaystyle A}

for the Jacobian matrix of ψ at y = γ(t). Now let {eu, ev} be an orthonormal basis in the coordinate directions of R2. Recognizing that the columns of

coordinate charts on the spacetime manifold, to the preferred, usually orthonormal basis on the tangent space, where calculations can be considerably simplified

Together with spacelike vectors, there are 6 classes in all. An orthonormal basis for Minkowski space necessarily consists of one timelike and three

_{i}{\sqrt {p_{i}}}|\phi _{i}\rangle \otimes |a_{i}\rangle } for some orthonormal basis { | a i ⟩ } i ⊂ H A {\displaystyle \{|a_{i}\rangle \}_{i}\subset {\mathcal

respect to the L 2 {\displaystyle L^{2}} inner product. It has an orthonormal basis of eigenfunctions { w k } k = 1 ∞ {\displaystyle \{w_{k}\}_{k=1}^{\infty

definition may appear counterintuitive at first, but consider an orthonormal basis e n {\displaystyle e_{n}} of an infinite-dimensional, separable Hilbert

operators of the quantum harmonic oscillator with respect to the above orthonormal basis is a † = ( 0 0 0 0 … 0 … 1 0 0 0 … 0 … 0 2 0 0 … 0 … 0 0 3 0 … 0 …

(\mathbf {i} ,\mathbf {j} ,\mathbf {k} )} is a positively oriented orthonormal basis of a three-dimensional oriented Euclidean vector space, while a 1

{\displaystyle \sigma (A)\subset \mathbb {R} } and there exists an orthonormal basis { v i } i ∈ N {\displaystyle \{v_{i}\}_{i\in \mathbb {N} }} of eigenvectors

quantities, it is useful to express the curvature tensor in an orthonormal basis. In an orthonormal basis of an observer the non-zero components in geometric units

_{B}^{x},} where { | x ⟩ } {\displaystyle \{|x\rangle \}} form an orthonormal basis. We would like to know what the agent can learn about the classical

between the two vectors. Two vectors are orthogonal if ⟨u, v⟩ = 0. An orthonormal basis is a basis where all basis vectors have length 1 and are orthogonal

, ( f k ′ ) k ′ {\displaystyle (e_{k})_{k},(f_{k'})_{k'}} be two orthonormal basis of the Hilbert spaces H 1 , H 2 {\displaystyle H_{1},H_{2}} , then

{\displaystyle V} is a blade that is the product of the members of an orthonormal basis for ⁠ W {\displaystyle W} ⁠. It can be shown that if I {\displaystyle

i } i = 1 ∞ {\displaystyle \{\beta _{i}\}_{i=1}^{\infty }} be an orthonormal basis of H {\displaystyle {\mathcal {H}}} , then { ϕ i } i = 1 ∞ = { α i

..., v2k be fixed, and let T denote their span. Then there is an orthonormal basis e1, ..., e2k of T with dual basis w1, ..., w2k such that ι ∗ ω = ∑

B=\{{\vec {w}}_{1},{\vec {w}}_{2},\dots ,{\vec {w}}_{n}\}} is an orthonormal basis for W {\displaystyle W} . Cornelius Lanczos (1988). Applied analysis

\nolimits _{x=1}^{n}p_{x}|x\rangle \langle x|} with respect to some orthonormal basis { | x ⟩ } x = 1 n {\displaystyle \{|x\rangle \}_{x=1}^{n}} . By writing

and e i {\displaystyle \mathbf {e} _{i}} are the components of an orthonormal basis in the current configuration. Also, ∇ ∘ v = ∑ i , j = 1 3 ∂ v i ∂

power iteration with a Gram–Schmidt process, to instead produce an orthonormal basis of these Krylov subspaces. Pick a random vector u 1 {\displaystyle

Jacobian. If the columns of the matrix Φ {\displaystyle \Phi } are an orthonormal basis for the null space of J = [ F x F λ ] {\displaystyle

are orthogonal to each other. For any normal matrix A, Cn has an orthonormal basis consisting of eigenvectors of A. The corresponding matrix of eigenvectors

{\displaystyle \Omega } . A construction can be done as follows: choose an orthonormal basis in H {\displaystyle {\mathcal {H}}} , let τ α ( x ) = x + α {\displaystyle

inner product over the real numbers onto itself. With respect to an orthonormal basis in such a space SO(4) is represented as the group of real 4th-order

non-trivially only on a subspace spanned by the first N of a fixed orthonormal basis {ei}, for some N, being the identity on the remaining basis vectors

classic monograph (Gakhov (1990)). Let en(θ) = einθ be the standard orthonormal basis of L2(T). Let H2(T) be Hardy space, the closed subspace spanned by

_{i=1}^{n}x_{i}x_{i}^{*}=I} if and only if { x i } {\displaystyle \{x_{i}\}} is an orthonormal basis), n > m {\displaystyle n>m} and { E i } {\displaystyle \{E_{i}\}}

where { | e k ⟩ } {\displaystyle \{\vert e_{k}\rangle \}} is a set of orthonormal basis vectors for Hilbert space, the Z k {\displaystyle Z_{k}} are complex

Eigenvalues of L, are real, positive, countable, unbounded (2) There is an orthonormal basis of L2(U) composed of eigenvectors of L. (See Spectral theorem.) Generates

method is to expand the predictor function X {\displaystyle X} in an orthonormal basis of L2 space, the Hilbert space of square integrable functions with

of normal operators. According to spectral theory, there exists an orthonormal basis {hk} that simultaneously diagonalizes S. This characterizes convolutions

translation. In matrix form, where the components are with respect to an orthonormal basis, [ x 1 ( X 1 , X 2 , X 3 , t ) x 2 ( X 1 , X 2 , X 3 , t ) x 3 ( X

is replaced by n/e at the ramified primes. Since characters are an orthonormal basis of the class functions, after showing some analytic properties of

way. Let w 1 , . . . , w m {\displaystyle w_{1},...,w_{m}} be an orthonormal basis (with respect to the usual inner product on R n {\displaystyle \mathbb

^{2}}}}={\frac {dt}{d\tau }}} The Minkowski metric tensor with an orthonormal basis for the (−+++) convention is η μ ν = ( − 1 0 0 0 0 1 0 0 0 0 1 0 0

Under these conditions, the ϕ n {\displaystyle \phi _{n}} form an orthonormal basis with respect to the scalar product ⟨ f , g ⟩ = ∫ a b f ( x ) g ( x

matrices involved in computing attitude are each composed of a TRIAD of orthonormal basis vectors. "TRIAD" derives its name from this observation. It is of

important property that a trace operation may be defined. Given an orthonormal basis { ψ n } {\displaystyle \{\psi _{n}\}} for the Hilbert space, the trace

to infinity. The corresponding normalised eigenfunctions form an orthonormal basis of H0. The k-th eigenvalue of D is given by the minimax principle

mirror cavities. To continue, a set of orthonormal modes forms an orthonormal basis which spans a modal space, or Hilbert space. The transformation from

This is equivalent to the assertion that every Hilbert space has an orthonormal basis. More generally, every closed subspace of a Hilbert space is complemented

in H under the iterates of T. Applying Gram–Schmidt one obtains an orthonormal basis (ei) for H. Let (Hi) be the corresponding nested sequence of "coordinate"

Σ {\displaystyle \Sigma } , { t a } {\displaystyle \{t^{a}\}} an orthonormal basis (with respect to the Killing form) of the Lie algebra of G {\displaystyle

operator. An orthonormal basis can be chosen so that T has 2 × 2 skew-symmetric matrices down the diagonal. Scaling the orthonormal basis, it follows that

above, when n = m {\displaystyle n=m} . Although it does not yield an orthonormal basis, an alternative normalization is sometimes preferred due to its simplicity:

v 1 , … , v n } {\displaystyle \{v_{1},\ldots ,v_{n}\}} forms an orthonormal basis of R n {\displaystyle \mathbf {R} ^{n}} . Let J {\displaystyle J}

to ∑ i v i ∗ A v i {\textstyle \sum _{i}v_{i}^{*}Av_{i}} given an orthonormal basis of V {\textstyle V} . Wielandt minimax formula (: 44 )—Let 1 ≤ i 1

density matrix is obtained from the density operator by a choice of an orthonormal basis in the underlying space. In practice, the terms density matrix and

possible with classical mechanics. In addition, the Bell states form an orthonormal basis and can therefore be defined with an appropriate measurement. Because

density matrix is obtained from the density operator by a choice of an orthonormal basis in the underlying space. In practice, the terms density matrix and

(where downsampling is omitted), and the Newland transform (where an orthonormal basis of wavelets is formed from appropriately constructed top-hat filters

ray of spinors mapping to one point on the Bloch sphere. Given an orthonormal basis, any pure state | ψ ⟩ {\displaystyle |\psi \rangle } of a two-level

{A}})~\left[{\boldsymbol {A}}^{-1}\right]^{\textsf {T}}~.} In an orthonormal basis, the components of A {\displaystyle {\boldsymbol {A}}} can be written

{\displaystyle f:D(A)\to H} . Typically, elementary spectral theory gives an orthonormal basis of  H {\displaystyle H} consisting of eigenvectors  v j {\displaystyle

spectral theorem for compact self-adjoint operators, there is an orthonormal basis un of H consisting of eigenvectors of A: A u n = μ n u n , {\displaystyle

} If K = C, the set of irreducible characters of G forms an orthonormal basis of Z(K[G]) with respect to the inner product ⟨ ∑ g ∈ G a g g , ∑ g

observable A has a pure point spectrum, it can be written in terms of an orthonormal basis of eigenvectors. That is, A has a spectral decomposition A = ∑ λ λ

functions β {\displaystyle \beta } and γ {\displaystyle \gamma } in an orthonormal basis. A functional multiple index model is given as below, with symbols

there principle eigenvector. For tensor fields a positively oriented orthonormal basis I ( x ) = ( I 1 ( x ) , I 2 ( x ) , I 3 ( x ) ) {\displaystyle I(x)=(I_{1}(x)

oscillator. The corresponding solutions provide an important choice of an orthonormal basis for L2(R) and are given by the "physicist's" Hermite functions. Equivalently

orthogonality of characters. The characters of G {\displaystyle G} form an orthonormal basis: 1 | H | ∑ h ∈ H χ g ( h ) χ g ′ ( h ) = { 1 g = g ′ 0 g ≠ g ′ {\displaystyle

gates and the controlled version of R k {\displaystyle R_{k}} : An orthonormal basis consists of the basis states | 0 ⟩ , … , | 2 n − 1 ⟩ . {\displaystyle

(j_{2}j_{5})j_{8})j_{9}m_{9}\rangle .} Both sets of functions provide a complete, orthonormal basis for the space with dimension ( 2 j 1 + 1 ) ( 2 j 2 + 1 ) ( 2 j 4 +

normal to their plane given by c = a × b. Given a set of right-handed orthonormal basis vectors { eℓ }, the cross product is expressed in terms of its components

The covariant derivative D μ ψ {\displaystyle D_{\mu }\psi } in an orthonormal basis uses the orthonormal frame { e μ } {\displaystyle \{e_{\mu }\}} to

dxdy.} F {\displaystyle {\mathcal {F}}} is a Hilbert space with orthonormal basis e n ( z ) = z n n ! , n ≥ 0. {\displaystyle e_{n}(z)={z^{n} \over

{R} ^{n},e^{-{\frac {1}{2}}x^{2}}d^{n}x)} to infinite dimension. An orthonormal basis in this Hilbert space, generalizing that of Hermite polynomials, is

bundle f *TN → M. In the equation, e1, ..., em is an arbitrary g-orthonormal basis of the tangent space to M and Rh is the Riemann curvature tensor,

is the Hall inner product, for which the Schur polynomials form an orthonormal basis. Similar to the ordinary Schur polynomials, there are numerous ways

where { e n } n = 1 ∞ {\displaystyle \{e_{n}\}_{n=1}^{\infty }} is an orthonormal basis. A theorem of A. Dvoretzky and C. A. Rogers asserts that every infinite-dimensional

eigenvectors | n ⟩ {\displaystyle |n\rangle } , which we assume to form an orthonormal basis for the Hilbert space of states H {\displaystyle {\mathfrak {H}}}

fixed Y {\displaystyle Y} and Z {\displaystyle Z} , then for any orthonormal basis v 1 , … , v n {\displaystyle v_{1},\ldots ,v_{n}} of the vector space

K} and the Peter–Weyl theorem asserts that the characters form an orthonormal basis for the space of square-integrable class functions on K {\displaystyle

representation of g {\displaystyle {\mathfrak {g}}} . If (Xs) is an orthonormal basis of g {\displaystyle {\mathfrak {g}}} for the Killing form, the

1996. I. Daubechies, S. Jaffard, and J.L. Journe, A simple Wilson orthonormal basis with exponential decay, SIAM J. Math. Anal., 22 (2), pp. 554–572,

it consists of those elements that commute with one-another. An orthonormal basis can be freely chosen on h {\displaystyle {\mathfrak {h}}} ; write

can be chosen to be the operators diagonal with respect to a given orthonormal basis of H . {\displaystyle H.} Since every finite-dimensional normed space

v_{n}} where w 1 , … , w d {\displaystyle w_{1},\ldots ,w_{d}} is any orthonormal basis of V {\displaystyle V} . (The sum is in fact independent of the choice

factors into the general formulae found in orthogonal coordinates. The orthonormal basis vectors for the μ , ν , ϕ {\displaystyle \mu ,\nu ,\phi } coordinate

of linear functionals. The theorem that every Hilbert space has an orthonormal basis. The Banach–Alaoglu theorem about compactness of sets of functionals

{\hat {\phi }}}=r\sin \theta \,{\boldsymbol {d\phi }}} represent an orthonormal basis, the Hodge dual of L {\displaystyle \mathbf {L} } is the one form

all eigenvalues of the matrix [aij(x)] are real, and there is an orthonormal basis of ℝn consisting of eigenvectors. Denote the eigenvalues by λi and

vectors | Φ i A ⟩ {\displaystyle |{\Phi _{i}^{A}}\rangle } that make an orthonormal basis in H A {\displaystyle H_{A}} and, correspondingly, vectors | Φ i B

{i} k\cdot x}}{\sqrt {|\Lambda |}}},} where χ1, ..., χq forms an orthonormal basis of ℂq. The allowed values of k ∈ ℝ3 are n/|Λ|1⁄3 with n ∈ ℤ3 +. For

2 {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2}} are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given

Pauli matrices, hence the σ {\displaystyle \sigma } notation. In an orthonormal basis with a standard spin frame, the conventional components are σ 0 α

{\displaystyle {\big (}e_{j}{\big )}_{j\in \mathbb {N} }} is the standard orthonormal basis in l 2 ( Z ) {\displaystyle l^{2}(\mathbb {Z} )} . Direct calculation

orthogonal, and there is no entanglement initially. Also, choose an orthonormal basis { | e i ⟩ } i {\displaystyle \{|e_{i}\rangle \}_{i}} for H A {\displaystyle

functional, and it is self-adjoint, and so one concludes that there is an orthonormal basis Y 1 , … , Y n {\displaystyle Y_{1},\ldots ,Y_{n}} of m {\displaystyle

{\displaystyle v,J(v)} is an oriented g p {\displaystyle g_{p}} -orthonormal basis of T p M . {\displaystyle T_{p}M.} A Lie group consists of a C∞ manifold

The dot product in Cartesian coordinates (Euclidean space with an orthonormal basis set) is simply the sum of the products of components. In orthogonal

} These four pure states are all maximally entangled and form an orthonormal basis of the Hilbert space of the two qubits.: 38–39 : 98  They provide

the vector space which has Q n , k {\displaystyle Q_{n,k}} as an orthonormal basis. We choose define a linear map τ : T L n ( d ) → V n , k {\displaystyle

identity for the standard Moore-Penrose inverse, which assumes an orthonormal basis in both spaces. In order to solve more general least-squares problems

Bell measurement by a measurement defined by a maximally entangled orthonormal basis. All possible such generalizations were discussed by Werner in 2001

representations that arise. The total angular momentum states form an orthonormal basis of V3: ⟨ J M | J ′ M ′ ⟩ = δ J , J ′ δ M , M ′   . {\displaystyle

} k = 0 d − 1 {\displaystyle \left\{k\right\}_{k=0}^{d-1}} be an orthonormal basis for C d {\displaystyle \mathbb {C} ^{d}} , and define ω = e 2 π i

query-efficient way to attack black-box image classifiers. Take a random orthonormal basis v 1 , v 2 , … , v d {\displaystyle v_{1},v_{2},\dots ,v_{d}} in R

/3} . The unit normal in the angular direction which completes the orthonormal basis can be calculated for θ s {\displaystyle \theta _{s}} and θ c {\displaystyle

2 {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2}} are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given

only possible measurement is between any two orthogonal states (an orthonormal basis). So, for example, measuring in the rectilinear basis gives a result

spherical basis used here. That is, there is a complete set of kets (i.e. orthonormal basis of joint eigenvectors labelled by quantum numbers that define the

and e i {\displaystyle \mathbf {e} _{i}} are the components of an orthonormal basis in the current configuration. The inner product is defined as A :

2 {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2}} are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given

{\displaystyle {\mathcal {A}}} with respect to a specifically chosen orthonormal basis B {\displaystyle B} with the coefficients given as the multidimensional

quantised field. The FW transformation is a unitary transformation of the orthonormal basis in which both the Hamiltonian and the state are represented. The eigenvalues

d i m ( Λ ) {\displaystyle \lambda =1,2,...,dim(\Lambda )} is an orthonormal basis for the Hilbert space associated with a quantum system Λ {\displaystyle

appropriate Sobolev space, the set of Chebyshev polynomials form an orthonormal basis, so that a function in the same space can, on −1 ≤ x ≤ 1, be expressed

above; the only difference is that now our frame is a anholonomic (orthonormal) basis on a curved spacetime, rather than a coordinate basis in flat spacetime

where ( w 1 , ⋯ , w k ) {\displaystyle (w_{1},\cdots ,w_{k})} is any orthonormal basis for w ⊂ R n {\displaystyle w\subset \mathbf {R} ^{n}} , viewed as

spectral theorem for compact self-adjoint operators, there is an orthonormal basis un of H consisting of eigenvectors of A: A u n = μ n u n , {\displaystyle

_{i}(x)} , where the { φ i } {\displaystyle \{\varphi _{i}\}} form an orthonormal basis of L 2 ( X ) {\displaystyle L_{2}(X)} . By the positivity of T K

(}\nabla (df){\big )}_{p}(e_{i},e_{i})} where e1, ..., em is any gp-orthonormal basis of TpM. From the perspective of local coordinates, as given above

non-zero eigenvalue; they are the internal coordinates and form an orthonormal basis for a (3N - 6)-dimensional subspace of the nuclear configuration space

geometric realization in Euclidean space, with vertices mapped to an orthonormal basis. Other metric structures are also used, involving length metrics obtained

a = a1e1 + a2e2, where a1 and a2 are real numbers, e1 and e2 are orthonormal basis vectors. The geometric product of two such vectors is a b = ( a 1

}n\rangle .} Given that the eigenstates of the Hamiltonian always form an orthonormal basis ⟨ m | n ⟩ = δ m n {\displaystyle \langle m|n\rangle =\delta _{mn}}

{i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right).} The orthonormal basis set for total angular momentum is | j , m ⟩ {\displaystyle |j,m\rangle

spaces have integer L0 eigenvalues constrains the choice of s: for an orthonormal basis xi, the vector 1/2 xi,12 + s2 must satisfy (s, λ) ∈ Z for all λ ∈

spectral theorem in Riemannian geometry states that there exists an orthonormal basis ϕ 0 , ϕ 1 , … , ϕ n , … {\displaystyle \phi _{0},\phi _{1},\ldots

This rule corresponds to the fact that the Z-spider represents an orthonormal basis - the computational basis. X-spider fusion See Z-spider fusion. Identity

overlapping inscribed 16-cells. Each 16-cell constitutes a distinct orthonormal basis for the choice of a coordinate reference frame. The 60 axes and 75

coordinate system is the Cartesian coordinate system, which consists of orthonormal basis vectors. This means that, a 1 = | a 1 | = a 2 = | a 2 | = ⋯ = a d

x 0 j , j = 1 , … , n {\displaystyle x_{0j},j=1,\dots ,n} form an orthonormal basis, for any right-hand side, we have obtained one solution therefore

r 0 {\displaystyle {\boldsymbol {r}}_{0}} and gradually builds an orthonormal basis { v 1 , v 2 , v 3 , … } {\displaystyle \{{\boldsymbol {v}}_{1},{\boldsymbol

{\displaystyle \chi _{S}} are known as Fourier characters, and they form an orthonormal basis for the space of all functions over { − 1 , 1 } n {\displaystyle \{-1

{\displaystyle \mathbf {R} } , in particular the matrix containing an orthonormal basis of the noise subspace, which is the orthogonal complement of a {\displaystyle

their corresponding vertices are 120° = √3 apart). A 16-cell is an orthonormal basis for a 4-dimensional coordinate system, because its 8 vertices define

0,} and Tf = 0 implies u = 0 and hence f = 0. There is a complete orthonormal basis  fn of L2(Ω) consisting of eigenfunctions of T. Thus T f n = μ n f

\quad \forall |v;n\rangle \in {\mathcal {F}}_{n}} . The standard orthonormal basis { | λ ⟩ } {\displaystyle \{|\lambda \rangle \}} for the zero fermionic

)\theta ^{-1}\,d\theta } are uniformly bounded. With respect to the orthonormal basis einθ convolution operators are diagonal and their operator norms are

powers of a act on F, so can be simultaneously diagonalized by an orthonormal basis ei. Since (ei)3 is a positive multiple of ei, rescaling if necessary

certain linear combinations of them called spin network states form an orthonormal basis. As spin network functions form a basis we can formally expand any