Variational principles for symplectic eigenvalues. (English) Zbl 1475.15021
Summary: If \(A\) is a real \(2n \times 2n\) positive definite matrix, then there exists a symplectic matrix \(M\) such that \(M^T AM=\operatorname{diag}(D, D)\), 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 \(d_{i+j-1}(A+B)\geqslant d_i(A)+d_j(B)\).
MSC:
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15A45 | Miscellaneous inequalities involving matrices |
References:
| [1] | Dutta, Arvind, B., Mukunda, N., and Simon, R., The real symplectic groups in quantum mechanics and optics. Pramana45(1995), 471-495. |
| [2] | 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 · Zbl 0863.15001 |
| [3] | Bhatia, R., Linear algebra to quantum cohomology: The story of Alfred Horn’s inequalities. Amer. Math. Monthly108(2001), 289-318. http://dx.doi.org/10.2307/2695237 · Zbl 1016.15014 |
| [4] | 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 · Zbl 1425.15015 |
| [5] | 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 · Zbl 1329.15048 |
| [6] | 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 · Zbl 1451.15012 |
| [7] | 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 · Zbl 1098.81004 |
| [8] | Hiroshima, T., Additivity and multiplicativity properties of some Gaussian channels for Gaussian inputs. Phys. Rev. A73(2006), 012330. |
| [9] | 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 · Zbl 1223.37001 |
| [10] | 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 · Zbl 1145.81019 |
| [11] | Jain, T. and Mishra, H. K., Derivatives of symplectic eigenvalues and a Lidskii type theorem. Preprint, 2020. arXiv:2004.11024 |
| [12] | 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 · Zbl 1292.81139 |
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