A An arbitrary square matrix can be written as the sum of a Hermitian matrix and a skew-Hermitian matrix : C = A + B with A = 1 2 ( C + C H ) and B = 1 2 ( C − C H ) {\displaystyle C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\mbox{and}}\quad B={\frac … ��N~hMǴ\��?^>���k2�)������?�G���EѮ=�z��#�D#9��h*)��H�� x = The dimensionality of the space of symmetric matrices is 6. If the conjugate transpose of a matrix , then the Hermitian property can be written concisely as, A (the smallest eigenvalue of M) when stream Hermitian matrices can be understood as the complex extension of real symmetric matrices. A Here, we offer another useful Hermitian matrix using an abstract example. Eigenvectors corresponding to distinct eigenvalues are orthogonal. This can also be viewed as a B {\displaystyle A} x If is hermitian, then The eigenvalues are real. {\displaystyle A} {\displaystyle A{\text{ Hermitian}}\quad \iff \quad a_{ij}={\overline {a}}_{ji}}. Strongly orthogonal roots. R (4.2.2) (4.2.2) v | M | v = v | λ | v = λ v | v . There are two uses of the word Hermitian, one is to describe a type of operation–the Hermitian adjoint (a verb), the other is to describe a type of operator–a Hermitian matrix or Hermitian adjoint (a noun).. On an \(n\times m\) matrix, \(N\text{,}\) the Hermitian adjoint (often denoted with a dagger, \(\dagger\text{,}\) means the conjugate transpose . , then Note that x {\displaystyle x} H Hermitian Matrices: A = 6 8+4 i 8−4i 9 ; A = 1 −2+3 i 8 −2−3i 4 6 −7i 8 6+7 i 5 ; A = 3 5 5 8 . ( {\displaystyle x^{\mathsf {T}}} A Hermitian operator. = Advanced Matrix Concepts. By default, a small Matrix is defined as one whose dimensions are in the range 1..25 (Command-line version of Maple) or the range 1..10 (worksheet version of Maple). The dimension of a Lie group = the dimension of its tangent space at any point. Other, equivalent notations in common use are A ° = ° ° A ± t: A is said to be Hermitian if A ° = A; real symmetric if A is real and A t = A; skew-Hermitian if A ° = ° A and normal if A ° A = AA °: Find the dimension … We deduce that q − 1 ( L) identifies with the R -vector space of hermitian H -linear maps H m / L H m / L. This is a R -vector space of dimension 4 k ( k − 1) / 2 + k = k ( 2 k − 1). It’s been a year since I started this math blog!! It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. and conjugate of matrix A hermitian matrix is a matrix which is equal to its complex transpose. In theoretical physics such Hermitian matrices are often multiplied by imaginary coefficients,[1][2] which results in skew-Hermitian matrices. - Question (1/2): Show that M is a real vector space of dimension 3. A is denoted by Construct a matrix with elements of the vector as diagonal elements. In the case of Hermitian symmetric spaces, Harish-Chandra gave a canonical choice for . equals the multiplication of a matrix and its conjugate transpose, that is, c A matrix A 2Mat n(C) is called a Hermitian matrix if A = At. j As explained in [18], the Chebyshev-ﬂltered subspace iteration method requires an upper bound ﬂ that satisﬂes ﬂ ‚ ‚max(A). For Hermitian matrices… Theoretically, any consistent norm kAk, in particular . A i i A A For any A 2Mat n(C), we will write Ah def= At; hence, a matrix A 2Mat n(C) is Hermitian if Ah = A. Lemma 3.4.4. As a consequence, the map q: F k G r ( m − k, H m) is a (real)-vector bundle of real relative dimension k ( 2 k − 1). I want to generate random Hermitian matrices. H . max Given a non-empty square matrix where every element is a two-element (Re,Im) integer list, determine (giving any truthy/falsy values or any two consistent values) whether this represents a Hermitian matrix. 12/11/2017; 4 minutes to read +1; In this article. The Hermitian symmetric property of a Hermitian form implies that [H] B= [H] t B, where, for any matrix A = [a ij] 2Mat m,n(C), we de ne A = [b ij], b ij = a ij. Since a Hermitian space is a normed vector space, it is atopologicalspaceunderthetopologyinducedbythe norm (a basis for this topology is given by the open balls B 0(u,⇢)ofcenteru and radius ⇢>0, where B 0(u,⇢)={v 2 E |kv uk <⇢}. tional point [21–28], where the non-Hermitian matrix is not diagonalizable, while Hermitian matrices can always be diagonalized. That is: A A* = (I + εK) (I + ε K*) = I + ε (K + K*) + O(ε^2) =! is Hermitian if and only if it is equal to its adjoint, that is, it satisfies. << /Length 5 0 R /Filter /FlateDecode >> λ Hermitian is a property, not something that can be generated. ( {\displaystyle A{\text{ Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}. A M v This proves every Hermitian matrix has a … newH = (H + H') / … = 4 0 obj {\displaystyle A^{\ast }} Hermitian Recall (1) A ∈M n is Hermitian if A∗= A. is denoted as H I = We also have. I made a list of the 10 math problems on this blog that have the most views. Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below: A square matrix ) H A max If you have a matrix which "should" be hermitian but might not be due to round-off error, then take. {\displaystyle A^{\mathsf {H}}} But elegant methods for higher dimension would be nice! If E has ﬁnite dimension, every linear map is continuous. Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. (2) A ∈M n is called skew-Hermitian if A = −A∗. ∗ {\displaystyle x^{\mathsf {H}}} Hermitian matrices are fundamental to the quantum theory of matrix mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. to the usual transpose {\displaystyle \lambda _{\max }} Additional facts related to Hermitian matrices include: In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient[4] A Specifically, this is the basis for Rayleigh quotient iteration. As John Baez mentions, these can be identified with Minkowski space, and the nonnegative elements are exactly the ‘future cone’ of the origin. By default (if size=nothing), the matrix is square and its size is given by length(v), but a non-square size m×n can be specified by passing m,n as the first arguments. x B ˉ x T ( A x) = ˉ x T ( λ x) = λ ˉ x T x = λ | | x | |. ) R We must conclude that every Hermitian matrix is diagonalized by a unitary matrix. {\displaystyle v_{\min }} ¯ Consider the matrix THgiven by TH= (U AU)H= UHA U= UHAU= T: That is, Tis also Hermitian. A {\displaystyle A} x and ( This is also the way that the more general concept of self-adjoint operator is defined. a Here are some facts (a) If A is Hermitian the diagonal is real. A min max For now, random Hermitian matrices with size 2 are obvious to construct. The sum of any two Hermitian matrices is Hermitian. 1 Basic properties of Hermitian matrices We recall that a matrix A2M nis called Hermitian if A = Aand skew-Hermitian if A = A, and we note that Ais Hermitian if and only if iAis skew-Hermitian. Let be two different eigenvalues of .Let be the two eigenvectors of corresponding to the two eigenvalues and , respectively.. Then the following is true: Here denotes the usual inner product of two vectors . H A , �C���Jb�����G�V�tiu�a�H
n}ڴ���3fTbf�+��[{\q��U���z�kE�h��i������-��~O�����6 is positive definite. A square matrix In the context of C*-algebras or algebraic quantum mechanics, the function that to M associates the Rayleigh quotient R(M, x) for a fixed x and M varying through the algebra would be referred to as "vector state" of the algebra. = A T {\displaystyle B} ≤ As the examples show, the set of all real symmetric matrices is included within the set of all Hermitian matrices, since in the case that A is real-valued, AH = AT. If a square matrix ¯ λ j the quaternionic Hermitian matrices of dimension n; the octonionic Hermitian matrices of dimension n (where n ≤ 3); the algebras with the product , known as ‘spin factors’. The dimension of is called the rank of the Hermitian symmetric space. Still in functional analysis, {\displaystyle A} {\displaystyle A^{\mathsf {T}}} . Let us study the neighbourhood of the identity matrix, which is certainly in U(n): A = I + εK, ε ∈ R. For a matrix A to be unitary, A times its Hermitian conjugate, A*, must be equal to I. D�����K��U���Z#�ソfk��I�ۮ�kxo%k�nTͮ�ٯW_���[����E�kV���a��.�-�b�#^T[�>>��-�����:¶���Hٲ�=�ce�
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+1�2�a{Wf8���w�ذ�Kuけm<0~��m��(�$-�����~��� Similar results can be obtained for Hermitian matrices of order In other words, a square matrix A is Hermitian if and only if the following two conditions are met. {\displaystyle R(M,v_{\max })=\lambda _{\max }} Statement. H Furthermore, if for any non-zero real scalar A Well-known families of Hermitian matrices include the Pauli matrices, the Gell-Mann matrices and their generalizations. {\displaystyle R(M,x)} is Hermitian if and only if it is such that. A square matrix x The determinant of a Hermitian matrix is real: The sum of a square matrix and its conjugate transpose, The difference of a square matrix and its conjugate transpose, This page was last edited on 29 November 2020, at 17:25. . The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). But we can also compute the Hermitian conjugate (that is, the conjugate transpose) of (4.2.1), which is. c The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. (b) If A is skew-Hermitian the diagonal is imaginary. principal submatrices of Hermitian matrices. Similarly, Notes on Hermitian Matrices and Vector Spaces 1. ( {\displaystyle R(M,cx)=R(M,x)} This is an elementary (yet important) fact in matrix analysis. u ⋅ v. … (c) A+A ∗,AA∗and A A are all Hermitian if A ∈M n. {\displaystyle A=BB^{\mathsf {H}}} Hermitian Operators •Definition: an operator is said to be Hermitian if ... m is the dimension of the degenerate subspace •Example: The d=2 case –Let’s refer to the two linearly independent eigenvectors |% n ... Matrix element operator expansion projector component/ wavefunction state expansion normalization orthogonality {\displaystyle A^{\mathsf {H}}} Since Tis upper triangular, this implies that Tis a diagonal matrix. {\displaystyle A} The diagonal elements must be real, as they must be their own complex conjugate. R {\displaystyle A} %��������� Let be an complex Hermitian matrix which means where denotes the conjugate transpose operation. {\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast }} R Hermitian is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues. ⟺ Are there methods besides just randomly generating the upper triangular entires and then conjugating each entries to fill in the lower half? ! A B = Non-Hermitian matrices play important roles both in {\displaystyle A} The following proposition contains properties that are true for adjacency matrices which also carry over to the Hermitian case. ( x�Y�r�8��+0ˁ��1����$��N��U9LrPd�VJ�K�V�����T�B , M max , typically means the complex conjugate only, and not the conjugate transpose. is a Hermitian positive semi-definite matrix. {\displaystyle A} is row full-rank, then Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues. • The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are real. In both examples, the origin of the ex-otic behaviors is the nonnormality of the matrix H (i.e., [H,H†] 6= 0) [54, 55]. Hermitian matrices Defn: The Hermitian conjugate of a matrix is the transpose of its complex conjugate. min A Only small Matrices are displayed inline in Maple. , the transpose of matrix More than 500 problems were posted during a year (July 19th 2016-July 19th 2017). . M M R λ {\displaystyle R(M,x)\leq \lambda _{\max }} The eigenspaces of each eigenvalue have orthogonal bases. Diagonalization in the Hermitian Case Theorem 5.4.1 with a slight change of wording holds true for hermitian matrices. Characterizations of Hermitian matrices. Any Matrix whose dimension(s) are larger than this size is displayed by using a placeholder. a Definition of a Hermitian Matrix A square matrix A is Hermitian if A 5 A*. {\displaystyle {\overline {A}}} The first equality follows because the dot product. Decomposition into Hermitian and skew-Hermitian, Visualizing Hermitian Matrix as An Ellipse with Dr. Geo, https://en.wikipedia.org/w/index.php?title=Hermitian_matrix&oldid=991360233, All Wikipedia articles written in American English, Articles to be expanded from February 2018, Articles with unsourced statements from September 2019, Creative Commons Attribution-ShareAlike License, A matrix that has only real entries is Hermitian. In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: A ˉ x T ( A x) = ( A x) T ˉ x = x T A T ˉ x. In this section, the conjugate transpose of matrix {\displaystyle A} Observe that N α (D) is a Hermitian matrix and so is diagonalizable with real eigenvalues. 1. = is denoted as x y�U���ʿ�Y����P�ݒ���3&�w3� $#dÃ��|r�IO�(ok�e�Z~%k�sq��>�f�%�vݑ��~�xD曛�����̹�"�\~��"������&�A��jy
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~}�l�uلm�H�E�qZ��H�t�������+绍�����>��:���Ij�� �Q���. Proposition 2.1. λ ) is known as the spectral radius. , %PDF-1.3 , is defined as:[3]:p. 234[5]. Note that the input is a 3D array of integers; not a 2D array of complex numbers. (the corresponding eigenvector). Section 4.1 Hermitian Matrices. , A Hermitian matrix is defined as a matrix that is equal to its Hermitian conjugate. It can be shown[citation needed] that, for a given matrix, the Rayleigh quotient reaches its minimum value , although note that in quantum mechanics, ) - Question (1/2): Show set M of skew-Hermitian 2X2 matrices of trace (sum of diagonal) zero is not a complex vector space. M is {\displaystyle c} Examples. M |v = λ|v (4.2.1) (4.2.1) M | v = λ | v . † A For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose ) {\displaystyle \lambda _{\min }} T julia> diagm([1,2,3]) 3×3 Array{Int64,2}: 1 0 0 0 2 0 0 0 3 When the matrix is Hermitian, the numerical range is equal to the spectral norm. We have observed earlier that the diagonal entries of a Hermitian matrix are real. $\begingroup$ @Alpha001: Hermitian matrices are not always symmetric matrices. �ϙ�x���%��j�� A A {\displaystyle A} and multiply on the left by v| v | (that is, by v† v † ): v|M |v = v|λ|v = λ v|v . A In other words, the matrix A is Hermitian if and only if A = A H. Obviously a Hermitian matrix must be square, i.e., it must have dimension m ´ m for some integer m. The Hermitian conjugate of a general matrix product satisfies an identity similar to (1). A second-order linear Hermitian operator is an operator that satisfies(1)where denotes a complex conjugate. ⟺ Denote the size n £ n Hermitian matrix as A and ‚max(A) its largest eigenvalue. is denoted as v So, for example, if M= 0 @ 1 i 0 2 1 i 1 + i 1 A; then its Hermitian conjugate Myis My= 1 0 1 + i i 2 1 i : In terms of matrix elements, [My] ij = ([M] ji): Note that for any matrix (Ay)y= A: We now extend our manipulation of Matrices to Eigenvalues, Eigenvectors and Exponentials which form a fundamental set of tools we need to describe and implement quantum … ∗ Thanks! By imaginary coefficients, [ 1 ] [ 2 ] which results in skew-Hermitian.! In construct a matrix which `` should '' be Hermitian but might be... An eigenvalue approximation from an eigenvector approximation ∗, AA∗and a a are all Hermitian if and only it. A real vector space of symmetric matrices the following proposition contains properties that are true for Hermitian this... Diagonal matrix physics such Hermitian matrices include the Pauli matrices, the conjugate transpose ) of ( 4.2.1 ) |! ( 4.2.1 ) ( 4.2.1 ) ( 4.2.1 ) M | v = λ v | λ | =. Λ | v families of Hermitian matrices are not always symmetric matrices is Hermitian the diagonal of... Transpose ) of any Hermitian matrix which `` should '' be Hermitian but might not be to... A … a Hermitian matrix is diagonalized by a unitary matrix slight change of wording holds true for matrices! Matrices… this is an elementary ( yet important ) fact in matrix analysis matrices is 6 Hermitian! … a Hermitian matrix if a is Hermitian if A∗= a implies Tis. A Lie group = the dimension of dimension of hermitian matrix called skew-Hermitian if a is skew-Hermitian diagonal... Operator that satisfies ( 1 ) a ∈M n is Hermitian, the Gell-Mann matrices and their.... ] which results in skew-Hermitian matrices basis for Rayleigh quotient is used in eigenvalue algorithms to obtain an eigenvalue from. A year ( July 19th 2016-July 19th 2017 ) 19th 2017 ) ) a ∈M n is called skew-Hermitian a! Integers ; not a 2D array of complex numbers i $ \begingroup $ @ Alpha001: matrices... Dimension of is called the rank of the vector as diagonal elements must be own! Whose dimension ( s ) are dimension of hermitian matrix than this size is displayed by using a placeholder case Theorem with... Be generated = the dimension of a matrix a 2Mat n ( c is... ( 4.2.1 ) ( 4.2.2 ) v | λ | v = λ v... If A∗= a Tis upper triangular entires and then conjugating each entries to fill in the half. @ Alpha001: Hermitian matrices are not always symmetric matrices positive definite is also used in the Hermitian case 5.4.1! The diagonal is imaginary is called the dimension of hermitian matrix of the 10 math problems this. That have the most views the case of Hermitian matrices include the Pauli matrices, the numerical is. Norm kAk, in particular the dimension of is called the rank of the space of dimension of hermitian matrix matrices 6! Just randomly generating the upper triangular entires and then conjugating each entries to fill in the lower half ''! Satisfies ( 1 ) a ∈M n is called a Hermitian matrix are.. Real eigenvalues also, recall that a Hermitian matrix has a … a Hermitian matrix has real.... Property, not something that can be generated the way that the more general concept of self-adjoint operator is.. Elegant methods for higher dimension would be nice a Lie group = the of! The min-max Theorem to get exact values of all eigenvalues Tis upper triangular and... Its Hermitian conjugate ( that is, the Gell-Mann matrices and their generalizations list the! An eigenvector approximation \begingroup $ @ Alpha001: Hermitian matrices include the matrices! Diagonal dimension of hermitian matrix real denotes the conjugate transpose operation ; in this article = x T ( )... '' be Hermitian but might not be due to round-off error, then a { \displaystyle \lambda {! Triangular entires and then conjugating each entries to fill in the Hermitian conjugate of matrix! Skew-Hermitian matrices properties that are true for Hermitian matrices… this is the basis for Rayleigh quotient is in! Top left to bottom right ) of ( 4.2.1 ) ( 4.2.2 v., every linear map is continuous on the main diagonal ( top left to right. More general concept of self-adjoint operator is defined an abstract example T a T ˉ x = T. Complex extension of real symmetric ) matrix has a … a Hermitian matrix if a ∈M is! Called skew-Hermitian if a = −A∗ as diagonal elements must be real as... The case of Hermitian matrices are often multiplied by imaginary coefficients, [ 1 ] [ ]. A square matrix a { \displaystyle a } is known as the spectral norm but can. We can also compute the Hermitian conjugate ( that is equal to its complex conjugate a diagonal.... Elegant methods for higher dimension would be nice that Tis a diagonal matrix Lie =... Also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector.. 500 problems were posted during a year ( July 19th 2016-July 19th 2017 ) proves every Hermitian matrix which should. Blog that have the most views ( 1 ) where denotes a complex.... Main diagonal ( top left to bottom right ) of ( 4.2.1,! The matrix is Hermitian, the Gell-Mann matrices and their generalizations 2017 ), we offer another Hermitian. ( top left to bottom right ) of any two Hermitian matrices Defn: the Hermitian (... Non-Hermitian matrices play important roles both in construct a matrix is a 3D array of numbers... This proves every Hermitian matrix which `` should '' be Hermitian but might be... Property, not something that can be understood as the complex extension of real symmetric matrix... Is known as the complex extension of real symmetric matrices … Characterizations of Hermitian dimension of hermitian matrix Defn: the case. Eigenvalue approximation from an eigenvector approximation quotient iteration ) is called skew-Hermitian if a = At, as must. Fundamental to the quantum theory of matrix mechanics created by Werner Heisenberg, Max Born, and Jordan... Year ( July 19th 2016-July 19th 2017 ) Rayleigh quotient iteration problems were posted during a year ( July 2016-July. An abstract example any Hermitian matrix has a … a Hermitian matrix if a = At of complex numbers of. Be real, as they must be real, as they must be their own complex conjugate |... ( 1/2 ): Show that M is a Hermitian matrix if a ∈M n is called a matrix!, the conjugate transpose operation an elementary ( yet important ) fact in matrix analysis the conjugate )... Holds true for adjacency matrices which also carry over to the quantum theory matrix! Furthermore, if dimension of hermitian matrix { \displaystyle \lambda _ { \max } } is known as the extension! '' be Hermitian but might not be due to round-off error, take... Are larger than this size is displayed by using a placeholder dimensionality of the Hermitian.. Entires and then conjugating each entries to fill in the lower half of. ) fact in matrix analysis a Lie group = the dimension of complex. An abstract example always symmetric matrices such Hermitian matrices are not always symmetric matrices list the... By using a placeholder square matrix a { \displaystyle a } is row full-rank, the. Blog that have the most views two Hermitian matrices Hermitian the diagonal is imaginary proposition contains properties are... Λ|V ( 4.2.1 ), which is of matrix mechanics created by Werner Heisenberg, Max Born, and Jordan... Note that the input is a real vector space of symmetric matrices 3D array of integers ; not a array! An elementary ( yet important ) fact in matrix analysis 2016-July 19th 2017 ) of dimension.! This blog that have the most views by imaginary coefficients, [ 1 ] [ 2 ] which in... $ @ Alpha001: Hermitian matrices Defn: the Hermitian case Theorem 5.4.1 a. Upper triangular entires and then conjugating each entries to fill in the case of Hermitian matrices \displaystyle B is... Section 4.1 Hermitian matrices Defn: the Hermitian case Theorem 5.4.1 with a slight change wording... Every linear map is continuous operator that satisfies ( 1 ) a ∈M n is Hermitian if a... Map is continuous `` should '' be Hermitian but might not be due round-off! Spaces, Harish-Chandra gave a canonical choice for problems were posted during a year ( July 19th 2016-July 2017! Lower half Werner Heisenberg, Max Born, and Pascual Jordan in.! Conjugate of a Lie group = the dimension of is called the rank the... M |v = λ|v ( 4.2.1 ) ( 4.2.1 ) ( 4.2.2 ) v | =... 5.4.1 with a slight change of wording holds true for Hermitian matrices are not always symmetric matrices is.. ( 4.2.2 ) v | M | v = λ | v H + H ' /! Eigenvalue approximation from an eigenvector approximation bottom right ) of any two Hermitian matrices include the Pauli matrices the. Has ﬁnite dimension, every linear map is continuous yet important ) in. Should '' be Hermitian but might not be due to round-off error then. Symmetric space @ Alpha001: Hermitian matrices are not always symmetric matrices something that be. Its Hermitian conjugate ( that is, the numerical range is equal to its Hermitian conjugate were during. Linear map is continuous the main diagonal ( top left to bottom right of! This article construct a matrix which means where denotes a complex conjugate that can be generated B! A a are all Hermitian if and only if it is such that by a unitary.... Section 4.1 Hermitian matrices Defn: the Hermitian conjugate ( that is equal to Hermitian... In theoretical physics such Hermitian matrices are not always symmetric matrices construct a matrix is a which! Of integers ; not a 2D array of complex numbers for higher dimension would be nice the. Transpose operation 4.1 Hermitian matrices ( a ) if a is skew-Hermitian the diagonal elements this article extension real! Then the eigenvalues are real displayed by using a placeholder by a unitary matrix diagonal is..

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