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Variational principles for symplectic eigenvalues

Part of: Variational principles of physics Basic linear algebra General theory of linear operators

Published online by Cambridge University Press:  20 August 2020

Rajendra Bhatia  and
Tanvi Jain
Rajendra Bhatia
Affiliation:
Ashoka University, SonepatHaryana131029, India e-mail: rajendra.bhatia@ashoka.edu.in
Tanvi Jain*
Affiliation:
Indian Statistical Institute, New Delhi110016, India
*
e-mail: tanvi@isid.ac.in
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Abstract

If A is a real $2n \times 2n$ positive definite matrix, then there exists a symplectic matrix M such that $M^TAM=\text {diag}(D, D),$ where D is a positive diagonal matrix with diagonal entries $d_1(A)\leqslant \cdots \leqslant d_n(A).$ We prove a maxmin principle for $d_k(A)$ akin to the classical Courant–Fisher–Weyl principle for Hermitian eigenvalues and use it to derive an analogue of the Weyl inequality $d_{i+j-1}(A+B)\geqslant d_i(A)+d_j(B).$

Keywords

Symplectic eigenvaluesmaxmin principleWeyl’s inequalities

MSC classification

Primary: 15A45: Miscellaneous inequalities involving matrices
Secondary: 47A75: Eigenvalue problems 49S05: Variational principles of physics (should also be assigned at least one other classification number in section 49)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 3 , September 2021 , pp. 553 - 559
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The work of RB is supported by a Bhatnagar Fellowship of the CSIR. TJ acknowledges financial support from SERB MATRICS grant number MTR/2018/000554.

References

Dutta, Arvind, B., Mukunda, N., and Simon, R., The real symplectic groups in quantum mechanics and optics . Pramana 45(1995), 471–495.Google Scholar
Bhatia, R., Matrix analysis . Graduate Texts in Mathematics, 169, Springer-Verlag, New York, 1997. http://dx.doi.org/10.1007/978-1-4612-0653-8 CrossRefGoogle Scholar
Bhatia, R., Linear algebra to quantum cohomology: The story of Alfred Horn’s inequalities . Amer. Math. Monthly 108(2001), 289–318. http://dx.doi.org/10.2307/2695237 CrossRefGoogle Scholar
Bhatia, R., Some inequalities for eigenvalues and symplectic eigenvalues of positive definite matrices . Int. J. Math. 30(2019), 1950055. http://dx.doi.org/10.1142/s0129167x19500551 CrossRefGoogle Scholar
Bhatia, R. and Jain, T., On symplectic eigenvalues of positive definite matrices . J. Math. Phys. 56(2015), 112201. http://dx.doi.org/10.1063/1.493582 CrossRefGoogle Scholar
Bhatia, R. and Jain, T., A Schur-Horn theorem for symplectic eigenvalues . Linear Algebra Appl. 599(2020), 133–139. http://dx.doi.org/10.1016/j.laa.2020.04.005 CrossRefGoogle Scholar
de Gosson, M., Symplectic geometry and quantum mechanics . Operator Theory: Advances and Applications, 166, Advances in Partial Differential Equations (Basel), Birkhäuser, Verlag, Basel, 2006. http://dx.doi.org/10.1007/3-7643-7575-2 CrossRefGoogle Scholar
Hiroshima, T., Additivity and multiplicativity properties of some Gaussian channels for Gaussian inputs . Phys. Rev. A 73(2006), 012330.CrossRefGoogle Scholar
Hofer, H. and Zehnder, E., Symplectic invariants and Hamiltonian dynamics . Modern Birkhäuser Classics, Birkhäuser Verlag, Basel, 2011. http://dx.doi.org/10.1007/978-3-0348-0104-1 CrossRefGoogle Scholar
Eisert, J., Tyc, T., Rudolph, T., Sanders, B. C., Gaussian quantum marginal problem . Comm. Math. Phys. 280(2008), 263–280. http://dx.doi.org/10.1007/s00220-008-0442-4 CrossRefGoogle Scholar
Jain, T. and Mishra, H. K., Derivatives of symplectic eigenvalues and a Lidskii type theorem. Preprint, 2020. arXiv:2004.11024 CrossRefGoogle Scholar
Krbek, M., Tyc, T., and Vlach, J., Inequalities for quantum marginal problems with continuous variables . J. Math. Phys. 55(2014), 062201–7. http://dx.doi.org/10.1063/1.4880198 CrossRefGoogle Scholar