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Longer titles found: Anti-diagonal matrix (view), Tridiagonal matrix (view)

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alternate case: diagonal matrix

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positive diagonal matrix D such that AD has all positive row sums. A has all positive diagonal elements, and there exists a positive diagonal matrix D such

is such that: A = LDU being L a unit lower triangular matrix, D a diagonal matrix and U a unit upper triangular matrix, then Doolittle's method produces

{\displaystyle D:=V^{T}AV} , we get a matrix whose top left block is the diagonal matrix λ I γ A ( λ ) {\displaystyle \lambda I_{\gamma _{A}(\lambda )}} . This

matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring R (whose identities are the zero 0 and one 1), where each

Laplacian matrix of the graph, defined as L = D − A, where D is the diagonal matrix of vertex degrees; the Laplacian nullity equals the cycle rank because

A square root of a 2×2 matrix M is another 2×2 matrix R such that M = R2, where R2 stands for the matrix product of R with itself. In general, there can

non-singular eigenvector matrix such that A = VΛV −1, where Λ is a diagonal matrix. If X ∈ Cn,n is invertible, its condition number in p-norm is denoted

A D = A − 1 {\displaystyle A^{\text{D}}=A^{-1}} . If A is a block diagonal matrix A = [ B 0 0 N ] {\displaystyle A={\begin{bmatrix}B&0\\0&N\end{bmatrix}}}

{\displaystyle Q^{-1}MQ} representing T {\displaystyle T} is a block-diagonal matrix Q − 1 M Q = [ A 0 0 B ] {\displaystyle Q^{-1}MQ=\left[{\begin{arra

{\displaystyle {\vec {F}}_{f}=-k{\vec {v}}} , where k {\displaystyle k} is a diagonal matrix, then the Rayleigh dissipation function can be defined for a system

as a zero in X {\displaystyle X} . Thus, a bloc-diagonal matrix is projected as a bloc-diagonal matrix and a triangular matrix is projected as a triangular

is a block diagonal matrix consisting of matrices of ones or results from row and/or column permutations of such a block diagonal matrix. Since the permanent

T}} are unitary matrices and Σ {\textstyle \mathbf {\Sigma } } is a diagonal matrix of size M × T {\textstyle M\times T} , that holds the singular values

{\displaystyle {\tilde {\lambda }}.} Let Λ {\displaystyle \Lambda } be the diagonal matrix having λ 1 , λ 2 , … , λ n {\displaystyle \lambda _{1},\lambda _{2}

2}/L_{1}&0&0\\0&M_{2}/M_{1}&0\\0&0&S_{2}/S_{1}\end{bmatrix}}} This diagonal matrix D maps cone responses, or colors, in one adaptation state to corresponding

equivalent to a diagonal matrix: A = V Λ V ∗ {\displaystyle A=V\Lambda V^{*}} for some unitary matrix V {\displaystyle V} and some diagonal matrix Λ   . {\displaystyle

ordering the configurations appropriately we may cast A as a block diagonal matrix: σ 1 ′ = + 1 σ 1 ′ = − 1 A = [ | A + | 0 | − − − | − − − | 0 | A −

{\displaystyle i=1,\dots ,n} ). Let Λ {\displaystyle \,\Lambda } be the diagonal matrix of the eigenvalues, i.e. Λ i j = λ i δ i j {\displaystyle \Lambda _{ij}=\lambda

disadvantage is that compact schemes are implicit and require to solve a diagonal matrix system for the evaluation of interpolations or derivatives at all grid

\mathbb {N} \}} basis set. This allows us to put the Hamiltonian in a diagonal matrix form where the diagonal elements are the only non-zero matrix elements

{\displaystyle L^{(\alpha )}=D^{-\alpha }LD^{-\alpha }\,} where D is a diagonal matrix and D i , i = ∑ j L i , j . {\displaystyle D_{i,i}=\sum _{j}L_{i,j}

The Frobenius complement H is the cyclic subgroup generated by the diagonal matrix whose i,i'th entry is zi. The Frobenius kernel K is the Sylow q-subgroup

where diag ⁡ ( A ) {\displaystyle \operatorname {diag} (A)} is the diagonal matrix with the same diagonal as A, and λ is a parameter that controls the

( M ) = n + − n − {\displaystyle \sigma (M)=n_{+}-n_{-}} where any diagonal matrix defining Q has n + {\displaystyle n_{+}} positive entries and n − {\displaystyle

the matrix [ I I i j ] {\displaystyle \left[I\!I_{ij}\right]} is a diagonal matrix, then they are called the principal directions. If the surface is oriented

{\displaystyle \rho (r_{i})W} . In this basis for V {\displaystyle V} , the diagonal matrix entry of ρ ( g ) {\displaystyle \rho (g)} vanishes for each basis vector

vectors u i j {\displaystyle u_{ij}} , S i {\displaystyle S_{i}} is the diagonal matrix holding the singular values s i j {\displaystyle s_{ij}} . For an n

{\displaystyle L=A^{-1}W} where A {\displaystyle A} is a positive diagonal matrix with entries A ( i , i ) {\displaystyle A(i,i)} corresponding to the

λi. The matrix S is defined up to multiplication on the right by a diagonal matrix. If A has no multiple eigenvalues, then let ci be the ith right eigenvector

= F−1ΛHF is from the definition of a circulant matrix, and ΛH is a diagonal matrix whose diagonal elements correspond to the first column of the circulant

operator L = D − A {\displaystyle L=D-A} , where D {\displaystyle D} is a diagonal matrix called the degree matrix and A {\displaystyle A} is the adjacency matrix

2 , e 3 {\displaystyle e_{1},e_{2},e_{3}} , the stress tensor is a diagonal matrix, and has only the three normal components λ 1 , λ 2 , λ 3 {\displaystyle

represented by the matrices obtained by swapping two adjacent rows of a diagonal matrix. The Weyl group is the symmetric group on n letters. More generally

TAT^{-1}} of A is as well a companion matrix of ƒ(X). Choosing T as diagonal matrix leaves the structure of A invariant. The root close to z k {\displaystyle

\Lambda _{X}} (resp. Λ Y {\displaystyle \Lambda _{Y}} ) denotes the diagonal matrix of the eigenvalues of Σ X X {\displaystyle \Sigma _{XX}} (resp. Σ Y

its diagonal, to transform H 0 {\displaystyle H_{0}} into a purely diagonal matrix D 0 = diag ⁡ ( d i ) {\displaystyle D_{0}=\operatorname {diag} (d_{i})}

off-diagonal matrix elements A α β {\displaystyle A_{\alpha \beta }} , with α ≠ β {\displaystyle \alpha \neq \beta } , are much smaller than the diagonal

d i a g ( x ) {\displaystyle \mathbf {diag} (\mathbf {x} )} is the diagonal matrix of vector x {\displaystyle \mathbf {x} } . N {\displaystyle \mathbf

sequence splits by taking any matrix with determinant −1, for example the diagonal matrix ( − 1 , 1 , … , 1 ) . {\displaystyle (-1,1,\dots ,1).} If n = 2 k +

{T} }z=(x+b)^{\mathrm {T} }D(x+b)+d^{\mathrm {T} }x+e,} where D is a diagonal matrix and where x represents a vector of uncorrelated standard normal random

a 1 , … , a n ) {\displaystyle d(a_{1},\ldots ,a_{n})} denotes the diagonal matrix with a 1 , … , a n {\displaystyle a_{1},\ldots ,a_{n}} on the diagonal

\mathrm {diag} \left(e^{i\theta },e^{-i\theta }\right)} denotes the diagonal matrix with diagonal entries e i θ {\displaystyle e^{i\theta }} and e − i

other words, the transition dipole moment can be viewed as an off-diagonal matrix element of the position operator, multiplied by the particle's charge

invertible then neither is PAP−1 or A. Conversely let D be the block-diagonal matrix corresponding to PAP−1, in other words PAP−1 with the asterisks zeroised

unitary matrices and Δ {\displaystyle \Delta } is the generalized diagonal matrix of the singular values (with the same shape as Z {\displaystyle Z}

slave robot. If the constant c {\displaystyle c} is replaced by a diagonal matrix C {\displaystyle \mathbf {C} } it is possible to adjust the compliance

other words, the transition dipole moment can be viewed as an off-diagonal matrix element of the position operator, multiplied by the particle's charge

_{1}^{0},\ldots ,\mathbf {R} _{N}^{0}} . The diagonal matrix M contains the masses on the diagonal. The diagonal matrix Φ {\displaystyle {\boldsymbol {\Phi }}}

Another method can extend Givens rotations to complex matrices. A diagonal matrix whose diagonal elements have unit magnitudes but arbitrary phases is

omitted because in confined systems such as atoms or molecules the diagonal matrix element ⟨ n | p x | n ⟩ = 0 {\displaystyle \langle n|p_{x}|n\rangle

{\displaystyle Q=E(I-A)(I+A)^{-1}} for some skew-symmetric matrix A and some diagonal matrix E with ±1 as entries. A slightly different form is also seen, requiring

In the Dirac-Pauli representation where β is a diagonal matrix, 5 is then reduced to a diagonal matrix: By elementary trigonometry, 6 also implies that:

the matrix of eigenvectors and D {\displaystyle \mathbf {D} } is the diagonal matrix of eigenvalues. The whitened data matrix is defined thus by X ← D −

_{j=1}^{3}P_{ij}(t){\frac {x_{j}(0)}{a_{j}(0)}}.} and we define a diagonal matrix A ( t ) {\displaystyle \mathbf {A} (t)} with diagonal elements being

with d = − b / a {\displaystyle d=-b/a} , we can get a diagonal matrix in B {\displaystyle B} . This shows the quotient group B / U ≅ F ×

blocks Aj (j = 1,...,p with p ≥ 2) are nonsingular. Define a block diagonal matrix D = diag(A1,...,Ap), then D is also nonsingular. Left-multiplying D−1

{\displaystyle \mathbf {Q} } is some K × K {\displaystyle K\times K} diagonal matrix. Typically, Q {\displaystyle \mathbf {Q} } is selected to be an identity

{\displaystyle \Delta _{i,i+2}=1/h_{i+1}} W is an (n-2)×(n-2) symmetric tri-diagonal matrix with elements: W i − 1 , i = W i , i − 1 = h i / 6 {\displaystyle W_{i-1

= S {\displaystyle DFT.DFT^{*}=S} , where S {\displaystyle S} is a diagonal matrix consisting of +1's and -1's. We can factor S = R R ∗ {\displaystyle

\limits _{j}w_{ij}} Also, let D be an n × n {\displaystyle n\times n} diagonal matrix with d {\displaystyle d} on the diagonal, and let W {\displaystyle

=-\mathbf {J} ^{\operatorname {T} }\mathbf {r} ,} where D is a positive diagonal matrix. Note that when D is the identity matrix I and λ → + ∞ {\displaystyle

g with a unitary transformation, it can be assumed that g(0) is a diagonal matrix with entries λi ≥ 0 with r = λ1 > 0 and that the image of the unit

matrix A {\displaystyle A} , S {\displaystyle S} is a (small) square diagonal matrix with non-negative singular values along the diagonal, and where V {\displaystyle

matrix A {\displaystyle A} , S {\displaystyle S} is a (small) square diagonal matrix with non-negative singular values along the diagonal, and where V {\displaystyle

memristor; its physical origin is the charge mobility in the conductor. The diagonal matrix and vector X = diag ⁡ ( X → ) {\displaystyle X=\operatorname {diag}

i ⟩ {\displaystyle |i\rangle } of the Hydrogen atom potential, the diagonal matrix elements go to zero, leaving us with i c ˙ 1 ( t ) = c 2 ( t ) cos

This is accomplished by multiplying the transformed diagonal matrix by another diagonal matrix D {\textstyle D} whose elements are also modulus of unity

&0\\0&0&0&1\end{matrix}}\right)} and the stress–energy tensor is a diagonal matrix T α β = ( ρ 0 0 0 0 p 0 0 0 0 p 0 0 0 0 p ) . {\displaystyle T^{\alpha

replacing the identity matrix ⁠ I {\displaystyle \mathbf {I} } ⁠ with the diagonal matrix consisting of the diagonal elements of ⁠ J T J {\displaystyle \mathbf

will be 1 {\displaystyle 1} s. The second term on the right will be a diagonal matrix with terms less than unity. The first term on the right is the "reduced

{\displaystyle \mathbf {A} \oplus \mathbf {B} } is defined as the block diagonal matrix of A {\displaystyle \mathbf {A} } and B {\displaystyle \mathbf {B}

Z2 symmetry generator T2 = UKUK = UU* = U (UT)−1 = Φ, where Φ is a diagonal matrix of phases. As a result, U = ΦUT and UT = UΦ, showing that U = Φ U Φ

that A = U D U ∗ {\textstyle A=UDU^{*}} , where D {\textstyle D} is a diagonal matrix containing the eigenvalues λ 1 , λ 2 , … , λ n {\textstyle \lambda

redistribution takes occurs on the path, the propagation matrix D is a diagonal matrix. The single entries consist of the Lambert-Beer absorption factor,

If we solve for all weights w {\displaystyle w} and put them in a diagonal matrix W {\displaystyle W} , stack all the new targets b ^ {\displaystyle

{\displaystyle \lambda _{i}} are the so-called dual variables forming the diagonal matrix Λ {\displaystyle \Lambda } . We can then provide an analytical expression

1,\ldots ,1)} where ξ a b {\displaystyle \xi _{ab}} is a constant diagonal matrix that can be expressed in terms of δ a b {\displaystyle \delta _{ab}}

{\displaystyle a_{ijs}} , L {\displaystyle \mathbf {L} } is a block diagonal matrix, and Π {\displaystyle \mathbf {\Pi } } is a permutation matrix regrouping

matrix can be written as a product of elementary matrices times a diagonal matrix; this follows easily from the fact that Z {\displaystyle \mathbb {Z}

{\displaystyle G} is walk-regular. A k {\displaystyle A^{k}} is a constant-diagonal matrix for all k ≥ 0. {\displaystyle k\geq 0.} Φ G − u ( x ) = Φ G − v ( x

{\displaystyle \Sigma \in {\mathcal {\textbf {R}}}^{D\times D}} is a diagonal matrix with elements { σ d 2 } d = 1 D {\displaystyle \left\{\sigma

unitary matrix, UT = U−1, of the eigenvectors ui of Γ and Λ is the diagonal matrix of eigenvalues λi. The frequency and shape of a mode is represented

=\langle \Psi _{2}|O_{x}|\Psi _{2}\rangle } More powerfully, the off-diagonal matrix elements between these distinct ground states for any local operator

and Δ {\displaystyle \Delta } is the n × n {\displaystyle n\times n} diagonal matrix, whose diagonal elements are δ i = ∑ d ∣ x i d ∤ x t x t < x i ( f

denotes the adjacency matrix of the graph and K {\displaystyle K} is the diagonal matrix with the outdegrees in the diagonal. The probability calculation is

methods, or color appearance models (CAMs), run a von Kries-style diagonal matrix transform in a slightly modified, LMS-like, space instead. They may

values of a matrix are its diagonal elements after it is converted to a diagonal matrix (that is, eigenvalues). Now, the characteristic values of the square

diagonal matrix with positive entries. Thus g = uaw with w = u* v, so that any K-biinvariant function on G corresponds to a function of the diagonal matrix

Let A ′ = V − 1 A V . {\displaystyle A'=V^{-1}AV.} Then A′ will be a diagonal matrix whose diagonal elements are eigenvalues of A. Replace each diagonal

observation noise is uncorrelated. That is, C Z {\displaystyle C_{Z}} is a diagonal matrix. In such cases, it is advantageous to consider the components of y

The full E8 Weyl group is generated by this subgroup and the block diagonal matrix H4⊕H4 where H4 is the Hadamard matrix H 4 = 1 2 [ 1 1 1 1 1 − 1 1 −

covariance matrix, T {\displaystyle T} is a target matrix (e.g., a diagonal matrix), and the shrinkage intensity λ ∈ ( 0 , 1 ) {\displaystyle \lambda

entangled state if the reduced state of each subsystem of ρ is the diagonal matrix ( 1 n ⋱ 1 n ) . {\displaystyle {\begin{pmatrix}{\frac {1}{n}}&&\\&\ddots

memristor; its physical origin is the charge mobility in the conductor. The diagonal matrix and vector X = diag ⁡ ( X → ) {\displaystyle X=\operatorname {diag}

entangled state if the reduced state of each subsystem of ρ is the diagonal matrix ( 1 n ⋱ 1 n ) . {\displaystyle {\begin{pmatrix}{\frac {1}{n}}&&\\&\ddots

squares problem). Note that, for symmetric matrices, a symmetric tri-diagonal matrix is actually achieved, resulting in the MINRES method. Because columns

direction of the optic axis of a uniaxial crystal), resulting in a diagonal matrix for the permittivity tensor ε: where the diagonal values are squares

g { k i } {\displaystyle {\bf {K}}={\rm {diag}}\,\{k_{i}\}} is the diagonal matrix of the degrees of the network's nodes. Then, for a network of classical

generated by X is not closed if and only if X is similar over C to a diagonal matrix with two entries of irrational ratio. Let h ⊂ g be a Lie subalgebra

orthogonal matrix Q, there is a choice of basis in which Q is a block diagonal matrix Q = diag ⁡ ( Q 1 , Q 2 , … , Q k ) , {\displaystyle Q=\operatorname

{\displaystyle (x_{i}-x)^{j}} ; W {\displaystyle \mathbf {W} } is a diagonal matrix of the smoothing weights w i ( x ) {\displaystyle w_{i}(x)} ; and y

{\displaystyle \mathbf {W} _{r}} be the N × N {\displaystyle N\times N} diagonal matrix with diagonal elements ( w r [ 0 ] , w r [ 1 ] , … , w r [ N − 1 ]

eigenbasis of A ^ {\displaystyle {\hat {A}}} is not a diagonal but a block diagonal matrix, i.e. the degenerate eigenvectors of A ^ {\displaystyle {\hat {A}}}

{L}}^{2}} . The modular T {\displaystyle T} -matrix is defined to be a diagonal matrix whose ( [ A ] , [ A ] ) {\displaystyle ([A],[A])} -entry is the θ {\displaystyle

{1}{\det(I-XA)}},} where I is the order n identity matrix and X is the diagonal matrix with diagonal [ x 1 , x 2 , … , x n ] . {\displaystyle [x_{1},x_{2}

covariance matrix of factor returns, and D {\displaystyle D} is a block diagonal matrix of specific returns. The matrix C {\displaystyle C} is then used for

}}=K\left({\bar {X}},{\bar {X}}\right)} and D {\displaystyle \mathbf {D} } is a diagonal matrix. Depending on the method and application various ways of selecting

in question. (It is represented in any coordinate system by a 3 × 3 diagonal matrix with equal values along the diagonal.) It is numerically equal to ⁠1/3⁠

relation is a diagonal matrix, the scoring function can not distinguish asymmetric facts. ComplEx: As DistMult uses a diagonal matrix to represent the

make a multi-touch capacitance touchscreen, is to sandwich an x/y or diagonal matrix of fine, insulation coated copper or tungsten wires between two layers

in first-order logic the cosmological constant the lambda baryon a diagonal matrix of eigenvalues in linear algebra a lattice molar conductivity in electrochemistry

diagonalisation and leveraging the property that the matrix exponential of a diagonal matrix is the same as element-wise exponentiation of its elements) exp ⁡ (

so that, in the matrix term, Ω {\displaystyle \Omega } is the block diagonal matrix with ΩI's on the diagonal. Then, since det ( I − t Ω 2 π i ) = det

D Q T , {\displaystyle (X^{*})^{T}X^{*}=Q\,D\,Q^{T},} where D is a diagonal matrix with positive entries (assuming X ∗ {\displaystyle X^{*}} has maximum

entangled state if the reduced state of each subsystem of ρ is the diagonal matrix [ 1 n ⋱ 1 n ] . {\displaystyle {\begin{bmatrix}{\frac {1}{n}}&&\\&\ddots

abelian subalgebra. Let ε1, ... εm be the basis of h∗ such that, for a diagonal matrix A, εk(ρA) is the kth diagonal entry of A. Clearly this is a basis for

disc containing only λi from the spectrum. The corresponding block-diagonal matrix ⨁ i T i {\displaystyle \bigoplus _{i}T_{i}} is the Jordan canonical

{\displaystyle \mathbf {\Sigma } } is a 3 × 3 {\displaystyle 3\times 3} diagonal matrix with Σ = ( s 0 0 0 s 0 0 0 0 ) {\displaystyle \mathbf {\Sigma }

_{1}+\lambda _{2}+\lambda _{3}} where Λ {\displaystyle \Lambda } is a diagonal matrix with eigenvalues λ 1 {\displaystyle \lambda _{1}} , λ 2 {\displaystyle

\eta _{ab}=\delta _{ab}} . For pseudo-Riemannian manifolds, it is the diagonal matrix having signature ( p , q ) {\displaystyle (p,q)} . The notation e i

{z} \quad {\text{(1.6)}}} where B {\displaystyle \mathbf {B} } is a diagonal matrix with the p {\displaystyle p} th diagonal element equal to ( α ( p )

package has an xij argument that allows the successive elements of the diagonal matrix to be inputted. Yee (2015) describes an R package implementation in

coefficients' freedom to vary. For example, if W {\displaystyle W} is the diagonal matrix of IRLS weights at convergence, and X {\displaystyle X} is the GAM

={\begin{pmatrix}1&1&0\end{pmatrix}}^{T}} and Ω {\displaystyle \Omega } is a diagonal matrix with entries Ω i i = σ i 2 {\displaystyle \Omega _{ii}=\sigma _{i}^{2}}

\operatorname {diag} } be the function that takes a vector and constructs a diagonal matrix whose on-diagonal entries are those of the vector. Let L − 1 {\displaystyle

can be written as B−J where J is the all-one matrix and B is a block-diagonal matrix whose diagonal blocks are of the form (n−3)I+4J. Moreover, he showed

} are orthogonal matrices and S {\displaystyle \mathbf {S} } is a diagonal matrix which contains the singular values of E e s t {\displaystyle \mathbf

package has an xij argument that allows the successive elements of the diagonal matrix to be inputted. Yee (2015) describes an R package implementation in

{\displaystyle Q} is a random orthogonal matrix and D {\displaystyle D} is a diagonal matrix with eigenvalues ranging from λ n = 1 {\displaystyle \lambda _{n}=1}

{for} \,\mathrm {all} \,t\right\}} . If η {\displaystyle \eta } is the diagonal matrix with diagonal entries (1, −1, −1, −1), then the Lie algebra o ( 1

1 {\displaystyle S''_{zz}(0)=PJ_{z}P^{-1}} , where Jz is an upper diagonal matrix containing the eigenvalues and det P ≠ 0; hence, det S z z ″ ( 0 )

matrix of ( K n , n ) {\displaystyle (K_{n,n})} looks like a block off-diagonal matrix, with two blocks of ones and two blocks of zeros. For example, the

{\displaystyle L=D-A} , where D {\displaystyle D} is the degree matrix (a diagonal matrix where D i i {\displaystyle D_{ii}} is the degree of vertex i) and A

classes. Left. S positive scalar times the identity matrix. Centre. D diagonal matrix with positive entries on the main diagonal. Right. F symmetric positive

Hr(A) there is an algebra automorphism carrying the matrix onto a diagonal matrix with real entries; it is then straightforward to check that [L(a),L(b)]

\Omega X)^{-1}\Omega {\vec {S}}} where X {\displaystyle X} is the diagonal matrix with elements x i {\displaystyle x_{i}} on the diagonal, α , β , χ

_{n}\psi _{n}^{*}(\theta ')\psi _{n}(\theta )} which represents a diagonal matrix representation in the basis of its plane-wave eigenfunctions ψ = exp

_{12}} (and similarly, σ 12 {\displaystyle \sigma _{12}} ) to be block diagonal matrix with blocks λ k ρ k {\displaystyle \lambda _{k}\rho _{k}} (and λ k

where U is a unit triangular matrix (with unit diagonal), and D is a diagonal matrix. Between the two, the U-D factorization uses the same amount of storage

and x j {\displaystyle x_{j}} . Define D {\displaystyle D} to be a diagonal matrix with D i i = ∑ j = 1 ℓ + u W i j {\displaystyle D_{ii}=\sum _{j=1}^{\ell

techniques or diagonal-loading SMI (where a small magnitude, random diagonal matrix is added to attempt to stabilize the matrix prior to inverting). This

are the principal stresses and the stress tensor is represented by a diagonal matrix: σ i j = [ σ 1 0 0 0 σ 2 0 0 0 σ 3 ] {\displaystyle \sigma

latter is obtained by taking the transpose and conjugating by the diagonal matrix with entries ±1. Hence H also contains ( a ¯ − c ¯ − b ¯ d ¯ ) . {\displaystyle

^{+}=\left\lceil E_{1}^{+},E_{2}^{+}\right\rfloor } for any block-diagonal matrix. Then define A i + = Q [ Y − 1 0 ] Σ i + U i ∗ {\displaystyle

{V}}D^{T}/n)^{-1}(D{\hat {\beta }}_{u}){\xrightarrow {d}}X_{Q}^{2}} Where D is a diagonal matrix with the last Q diagonal entries as 0 and others as 1. If the restricted

to take advantage of economies of scale. Firms operating along the diagonal matrix are assumed to perform better than those too far from the diagonal

of C {\displaystyle C} remark first that this is the case for any diagonal matrix C {\displaystyle C} , because the coordinate-wise maximizer is independent

axis fixed (Euler's rotation theorem), and is conjugate to a block diagonal matrix of the form D = ( cos ⁡ θ − sin ⁡ θ 0 sin ⁡ θ cos ⁡ θ 0 0 0 1 ) = e

radial basis function for each neuron and C {\displaystyle C} is a diagonal matrix which contains elements that have been set during training and correspond

_{k}(\mathbb {C} )} , and the Schur polynomial is also the character of the diagonal matrix d i a g ( x 1 , … , x k ) {\displaystyle \mathrm {diag} (x_{1},\ldots

matrix with elements Qil, and diag(n − 1, n − 2, ..., 1, 0) means the diagonal matrix with the elements n − 1, n − 2, ..., 1, 0 on the diagonal. See proposition

{\begin{pmatrix}0&M\\M^{T}&T\end{pmatrix}}-m,} where T {\displaystyle T} is a block diagonal matrix whose blocks are 2 × 2 {\displaystyle 2\times 2} submatrices of the

{\alpha }}_{1}^{-1},\dots ,{\boldsymbol {\alpha }}_{N}^{-1})} is a block-diagonal matrix with each block α j − 1 ∈ C 3 × 3 {\displaystyle {\boldsymbol {\alpha

the case of H = SU(2) the symmetry σ is given by conjugation by the diagonal matrix with entries ±i so that σ ( α β − β ¯ α ¯ ) = ( α − β β ¯ α ¯ ) {\displaystyle

exposure probabilities for each unit in the analysis. First, define a diagonal matrix with a vector of treatment assignment probabilities P = diag ⁡ ( p

spectral subspace corresponding to the interval [a,b]. Let g(s) be the diagonal matrix with entries s and s−1 for |s| > 1. Then g(s)P[a,b]g(s)−1 = P[s2a,

{x}}_{T}^{-1}.\end{aligned}}} Here, v ^ {\displaystyle {\hat {v}}} refers to a diagonal matrix where the ( i , i ) {\displaystyle (i,i)} element is the i {\displaystyle

Since each parameter only affects one reaction, the matrix will be a diagonal matrix: E = [ ∂ v 1 ∂ k 1 0 0 0 0 ∂ v 2 ∂ k 2 0 0 0 0 ∂ v 3 ∂ k 3 0 0 0 ∂

the basis, corresponding to fixed number of particles, it is a non-diagonal matrix. Its eigenvector—the state vector of a physical system—is an infinite

{T}}} ⁠, where Q {\displaystyle Q} is orthogonal and S is a block diagonal matrix with ⌊ n / 2 ⌋ {\textstyle \lfloor n/2\rfloor } blocks of order 2,

{\displaystyle A_{g}={\mbox{diag}}\{A_{1,g},\ldots ,A_{d,g}\}} is a diagonal matrix whose elements are proportional to the eigenvalues of Σ g {\displaystyle

corresponds to the automorphism of E = Hp + q(R) given by conjugating by the diagonal matrix with p diagonal entries equal to 1 and q to −1. Without loss of generality

+ |λ|2)−(k−j)/2. It is therefore compact because it is given by a diagonal matrix with diagonal entries tending to zero. Elliptic regularity (Weyl's

{\displaystyle dp_{1},\dots ,dp_{n-1}} from which we form the square diagonal matrix D {\displaystyle \mathbf {D} } , the columns of which span the rectangle

matrix Σ indicates which space directions are restricted and is a diagonal matrix consisting of zeros and ones. Which spatial direction is restricted

left eigenstates. This construction can be extended to include off-diagonal matrix elements and gauge projections. Physical quantities can then be computed

representing g in Vj is just S2j ( g ). Since every g is conjugate to a diagonal matrix with diagonal entries ζ {\displaystyle \zeta } and ζ − 1 {\displaystyle

{\displaystyle m\gtrsim 2s} in a large photon-rich system, the off-diagonal matrix elements in H ^ T C {\displaystyle {\hat {H}}_{TC}} (above) replace