On the diagonalizability of a matrix by a symplectic equivalence, similarity or congruence transformation

Cruz, Ralph John de la; Faßbender, Heike

doi:10.1016/j.laa.2016.01.030

Cited by 16 publications

(9 citation statements)

References 10 publications

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“…The proof we present is found in [HJ13, Cor. 4.4.4] for the complex case, and in [dlCF16] for the symplectic case. For the sake of the reader, we give a unified proof which employs also tools from [IF05] in the symplectic case.…”

Section: Appendix a Matrix Factorizationsmentioning

confidence: 99%

Symmetric Fermi projections and Kitaev's table: topological phases of matter in low dimensions

Gontier,

Monaco,

Perrin-Roussel

2022

Preprint

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We review Kitaev's celebrated "periodic table" for topological phases of condensed matter, which identifies ground states (Fermi projections) of gapped periodic quantum systems up to continuous deformations. We study families of projections which depend on a periodic crystal momentum and respect the symmetries that characterize the various classes of topological insulators. Our aim is to classify such families in a systematic, explicit, and constructive way: we identify numerical indices for all symmetry classes and provide algorithms to deform families of projections whose indices agree. Aiming at simplicity, we illustrate the method for 0-and 1-dimensional systems, and recover the (weak and strong) topological invariants proposed by Kitaev and others.(T -symmetry).If ε T = 1, this T -symmetry is said to even, and if ε T = −1 it is odd.Definition 1.2 (Charge-conjugation symmetry). Let C : H → H be an antiunitary operator such thatWe say that a continuous map P :If ε C = 1, this C-symmetry is said to even, and if ε C = −1 it is odd.Definition 1.3 (Chiral symmetry). Let S : H → H be a unitary operator such that S 2 = I H . We say that P : T d → G n (H) satisfies chiral or sublattice symmetry, or in short S-symmetry, ifThe simultaneous presence of two symmetries implies the presence of the third. In fact, the following assumption is often postulated [RSFL10]: Assumption 1.4. If the Hilbert space H is endowed with anti-unitary operators T and C as in Definitions 1.1 and 1.2 respectively, then we assume that their product S := T C is an operator as in Definition 1.3, that is, S is unitary and S 2 = I H .Remark 1.5. This assumption is tantamount to require that the operators T and C commute or anti-commute among each other, depending on their even/odd nature. Indeed, the product of two anti-unitary operators is unitary, and the requirement that T C behaves as in Definition 1.3 readsThe same sign determines whether S commutes or anti-commutes with T and C.

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Section: Appendix a Matrix Factorizationsmentioning

confidence: 99%

Symmetric Fermi projections and Kitaev's table: topological phases of matter in low dimensions

Gontier,

Monaco,

Perrin-Roussel

2022

Preprint

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“…So by [dlCb16,Theorem 21] we get that M is symplectically congruent to a diagonal matrix. Now, consider an element M P X 2r .…”

Section: Complete Symmetric Symplectic Formsmentioning

confidence: 99%

Complete symplectic quadrics and Kontsevich spaces of conics in Lagrangian Grassmannians

Corniani¹,

Massarenti²

2020

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A wonderful compactification of an orbit under the action of a semi-simple and simply connected group is a smooth projective variety containing the orbit as a dense open subset, and where the added boundary divisor is simple normal crossing. We construct the wonderful compactification of the space of symmetric and symplectic matrices, and investigate its geometry. As an application, we describe the birational geometry of the Kontsevich spaces parametrizing conics in Lagrangian Grassmannians. Contents 1. Introduction 1 2. Complete quadrics 4 3. Complete symmetric symplectic forms 8 4. Divisors on S 2r 14 5. Birational geometry of S 2r 17 6. Moduli spaces of conics in Lagrangian Grassmannians 20 References 28

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“…Note that in the example above the corresponding quadratic representant is not diagonal, whereas the Hamiltonian H obviously is. The criterium for symplectic diagonalisability is presented in [8], where it is shown that a diagonalizable matrix A is simplecticaly diagonalizable if and only if it satisfies AJ 0 A T J 0 = J 0 A T J 0 A.…”

Section: Symplectic Hyperboloidsmentioning

confidence: 99%

Rabinowitz Floer homology for tentacular Hamiltonians

Pasquotto,

Vandervorst,

Wiśniewska

2018

Preprint

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This paper extends the definition of Rabinowitz Floer homology to noncompact hypersurfaces. We present a general framework for the construction of Rabinowitz Floer homology in the non-compact setting under suitable compactness assumptions on the periodic orbits and the moduli spaces of Floer trajectories. We introduce a class of hypersurfaces arising as the level sets of specific Hamiltonians: strongly tentacular Hamiltonians for which the compactness conditions are satisfied, cf. [19], thus enabling us to define the Rabinowitz Floer homology for this class. Rabinowitz Floer homology in turn serves as a tool to address the Weinstein conjecture and establish existence of closed characteristics for non-compact contact manifolds.

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On the diagonalizability of a matrix by a symplectic equivalence, similarity or congruence transformation

Cited by 16 publications

References 10 publications

Symmetric Fermi projections and Kitaev's table: topological phases of matter in low dimensions

Symmetric Fermi projections and Kitaev's table: topological phases of matter in low dimensions

Complete symplectic quadrics and Kontsevich spaces of conics in Lagrangian Grassmannians

Rabinowitz Floer homology for tentacular Hamiltonians

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