J. von Neumann. Some matrix-inequalities and metrization of matricspace. Tomsk Univ. Rev., 1:205--218, 1937.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....this survey, I would like to suggest its avour with a celebrated classical result. This result, von Neumann s characterization of unitarily invariant matrix norms, serves both as a historical jumping o point and as an elegant juxtaposition of the central ingredients of this article. Von Neumann [64] was interested in unitarily invariant norms k k on the vector space M of n by n complex matrices: kUXV k = kXk for all U; V 2 U ; X 2 M where U denotes the group of unitary matrices. The singular value decomposition shows that the invariants of a matrix X under unitary ....

J. von Neumann. Some matrix inequalities and metrization of matric-space. Tomsk University Review, 1:286-300, 1937. In: Collected Works, Pergamon, Oxford, 1962, Volume IV, 205-218.

....through rigid rotations or translations of all the N atoms are considered equivalent. The cRMSD between two equivalence classes # is defined as the minimal cRMSD with respect to all possible translations and rotations of the two structures. This minimization can be performed analytically [26, 27]. It has been noted that the cRMSD tends to increase with the number of atoms, N . Maiorov and Crippen proposed a normalization meant to make the cRMSD e#ectively independent of N [28] This correction is important if one compares distances corresponding to di#erent N . However, in the present ....

von Neumann J. Some matrix-inequalities and metrization of matric-space. Tomsk Univ Rev 1937; 1: 286--300.

....on isospectral optimality for di#erent unitarilyinvariant norms: a matrix norm is unitarily invariant if #PXQ# for every P , Q SO(n) X M n,n [HJ91, p. 203] We commence Section 2 with von Neumann s celebrated characterization of unitarily invariant norms in terms of symmetric gauges [vN37]. The most important family of unitarily invariant norms are the so called Schatten p norms, 1 # #. In the remainder of the section we determine the range of p such that a Schatten p norm is isospectrally optimal. We remark that the derivative of the Frobenius norm h(t) F along the flow ....

....of singular values of X: to ensure proper definition, we assume that the elements of sing(X) are arranged in increasing order, but this will play no further role in our discussion. Note that for Sym(n) the map sing(X) produces the absolute values of the eigenvalues of the matrix. Theorem 1 ([vN37]) A matrix norm is unitarily invariant if and only if there exists a symmetric gauge = sing(X) X M n,n . 2.1) We have already mentioned that the standard # p [R ] norm, 1 is a symmetric gauge. Therefore, by Theorem 1, it gives rise to a unitarily invariant norm, the Schatten ....

J. von Neumann, Some matrix inequalities and metrization of matrix space, Tomsk University Review 1 (

.... Sigma. This result is due to Ky Fan [12] The linear functional is minimized when trace(Q T SigmaA) is maximized. The maximum of this trace is given by the sum of the singular values of the matrix Sigma T A. This upper bound on the trace functional has been established by J. Von Neumann in [16], see also [11] Separate minimization of the quadratic and the linear part are well understood methods. The analytical solution of the orthogonal Procrustes problem for Stiefel matrices is to the best of our knowledge an open problem. It will be useful to interpret the minimization (1.4) ....

J. Von Neumann, Some matrix inequalities and metrization of the matrix space, Tomsk Univ. Rev. (1937), 286--300.

....1. Introduction. A norm # # : C pq # R is said to be unitarily invariant if for any A # C pq , #UAV # = #A# for all U # U(p) V # U(q) where U(p) denotes the group of p p unitary matrices. The characterization of unitarily invariant norms is well known and is due to von Neumann [20, 2]. Among the unitarily invariant norms, Ky Fan s k norms, # # k , are the most important ones due to the following result of Ky Fan [2] The Ky Fan k norms # # k : C pp # R, defined by #A# k = # k i=1 s i (A) where s 1 (A) # # s n (A) are the singular values of A. Our main purpose in ....

....# . For definiteness we assume p # q. Let G be the group of action of U(p) U(q) on C pq defined by A ## UAV # , where U # U(p) V # U(q) By the singular value decomposition, for any A # C pq , there exist U # U(p) V # U(q) such that A = U#V . A well known result of von Neumann [20] asserts that max Re tr AUBV : U # U(p) V # U(q) p # i=1 s i (A)s i (B) where s 1 (A) # s 2 (A) # # s p (A) are the singular values of A. Thus (C pq , G, F ) is an Eaton triple with reduced triple (W, H,F ) where W is the space p q real diagonal matrices and F is the cone ....

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J. von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (

....) j X i q p (1) i p (2) i : 2.64) Moreover, B(p (1) p (2) F (ae 1 ; ae 2 ) 2.65) since, given the eigenvalues of both density operators, the fidelity is maximized by choosing their eigenvectors to be the same, assigned to eigenvalues in order of size. This follows easily from [23], 24] and the representation of the square root of the fidelity as max unitary U trae 1=2 1 ae 1=2 2 U : 2.66) This completes the proof of the lemma. Now consider the situation where d is the dimension of each of two spaces Q and R, and ae RQ a density operator on the d 2 ....

J. von Neumann, "Some matrix-inequalities and metrization of matrix-space," Tomsk. Univ. Rev., vol. 1, pp. 286, 1937.

....(2) X i q p (1) i p (2) i : 45) Moreover, B(p (1) p (2) F ( 1 ; 2 ) 46) since, given the eigenvalues of both density operators, the delity is maximized by choosing their eigenvectors to be the same, assigned to eigenvalues in order of size. This follows easily from [39], 40] and the representation of the square root of the delity as max unitary U tr 1=2 1 1=2 2 U : 47) This completes the proof of the lemma. Now consider the situation where d is the dimension of each of two spaces Q and R, and RQ a density operator on the d 2 dimensional ....

J. von Neumann, \Some matrix-inequalities and metrization of matrix-space," Tomsk. Univ. Rev., vol. 1, pp. 286, 1937.

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J. von Neumann. Some matrix-inequalities and metrization of matricspace. Tomsk Univ. Rev., 1:205--218, 1937.

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