Two-sided hyperbolic SVD

Šego, Vedran

doi:10.1016/j.laa.2010.06.024

Cited by 9 publications

(6 citation statements)

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“…Due to the wide range of applications of spaces with hyperbolic inner product, the signature matrices are applied so far in the scientific literature in constructing the indefinite inner product space (see, for example, [9,[11][12][13]15,24,30]). Note that the signature matrix M ¼ diagðAE1Þ is both Hermitian and orthogonal, i.e.…”

Section: Choice Of Appropriate Metric Matrix Mmentioning

confidence: 99%

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Representations and computations of{2,3∼}and{<

Petrović

Stanimirović

2015

Applied Mathematics and Computation

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Section: Choice Of Appropriate Metric Matrix Mmentioning

confidence: 99%

“…The indefinite conjugate transpose matrix A $ , as well as the hyperbolic conjugate transpose matrix particulary, possess similar property with A Ã [24]: ðAx; yÞ G 2 ¼ ðx; A $ yÞ G 1 ;…”

Section: Introductionmentioning

confidence: 99%

Representations and computations of{2,3∼}and{<

Petrović

Stanimirović

2015

Applied Mathematics and Computation

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“…For our choice, eq. ( 7), we can resort to the hyperbolic singular value decomposition (SVD) [30] (Theorem 3.4, p. 1268) which under fairly general conditions grants us the existence of a generalized Schmidt decomposition (by virtue of the close connection between the Schmidt decomposition and the SVD, see, e.g., [31], Sec. 2.5, p. 109).…”

Section: The Generalized Schmidt Decompositionmentioning

confidence: 99%

“…2.5, p. 109). It should be mentioned here that theorem 3.4 in [30] provides us only with real coefficients in the (generalized) Schmidt decomposition. But taking into account the fact that in our consideration the spaces K A , K B are independent of each other we can always choose the coefficients to be non-negative (by means of a reflection).…”

Section: The Generalized Schmidt Decompositionmentioning

confidence: 99%

Entanglement capabilities of the spin representation of (3+1)D-conformal transformations

Scharnhorst

2011

Preprint

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Relying on a mathematical analogy of the pure states of the two-qubit system of quantum information theory with four-component spinors we introduce the concept of the intrinsic entanglement of spinors. To explore its physical sense we study the entanglement capabilities of the spin representation of (pseudo-) conformal transformations in (3+1)-dimensional Minkowski space-time. We find that only those tensor product structures can sensibly be introduced in spinor space for which a given spinor is not entangled.

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“…There is plenty of research on the subject, i.e., hyperbolic SVD [17,24], J 1 J 2 -SVD [9], two-sided hyperbolic SVD [20], hyperbolic CS decomposition [8,10] and indefinite QR factorization [19].…”

Section: Introductionmentioning

confidence: 99%

The hyperbolic Schur decomposition

Šego

2014

Linear Algebra and its Applications

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We propose a hyperbolic counterpart of the Schur decomposition, with the emphasis on the preservation of structures related to some given hyperbolic scalar product. We give results regarding the existence of such a decomposition and research the properties of its block triangular factor for various structured matrices.Keywords: indefinite scalar products, hyperbolic scalar products, Schur decomposition, Jordan decomposition, quasitriangularization, quasidiagonalization, structured matrices 2000 MSC: 15A63, 46C20, 65F25 IntroductionThe Schur decomposition A = U T U * , sometimes also called Schur's unitary triangularization, is a unitary similarity between any given square matrix A ∈ C n×n and some upper triangular matrix T ∈ C n×n . Such a decomposition has a structured form for various structured matrices, i.e., T is diagonal if and only if A is normal, real diagonal if and only if A is Hermitian, positive (nonnegative) real diagonal if and only if A is positive (semi)definite and so on.Furthermore, the Schur decomposition can be computed in a numerically stable way, making it a good choice for calculating the eigenvalues of A (which are the diagonal elements of T ) as well as the various matrix functions (for more details, see [11]). Its structure preserving property allows to save time and memory when working with structured matrices. For example, computing the value of some function of a Hermitian matrix is reduced to working with a diagonal matrix, which involves only evaluation of the diagonal elements.Unitary matrices are very useful when working with the traditional Euclidean scalar product x, y = y * x, as their columns form an orthonormal basis of C n . However, many applications require a nonstandard scalar product which is usually defined by [x, y] J = y * Jx, where J is some nonsingular matrix, and many of these applications consider Hermitian or skew-Hermitian J. The hyperbolic Email address: vsego@math.hr (VedranŠego) scalar product defined by a signature matrix J = diag(j 1 , . . . , j n ) (j k ∈ {−1, 1}) arises frequently in applications. It is used, for example, in the theory of relativity and in the research of the polarized light. More on the applications of such products can be found in [10,13,14,17].The Euclidean matrix decompositions have some nice structure preserving properties even in nonstandard scalar products, as shown by Mackey, Mackey and Tisseur [16], but it is often worth looking into versions of such decompositions that respect the structures related with the given scalar product. There is plenty of research on the subject, i.e., hyperbolic SVD [17,24], J 1 J 2 -SVD [9], two-sided hyperbolic SVD [20], hyperbolic CS decomposition [8,10] and indefinite QR factorization [19].There are many advantages of using decompositions related to some specific, nonstandard scalar product, as such decompositions preserve structures related to a given scalar product. They can simplify calculation and provide a better insight into the structures of such structured matrices.In this paper w...

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Two-sided hyperbolic SVD

Cited by 9 publications

References 8 publications

Representations and computations of{2,3∼}and{<

Representations and computations of{2,3∼}and{<

Entanglement capabilities of the spin representation of (3+1)D-conformal transformations

The hyperbolic Schur decomposition

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