Geodesic
Derivatives of symplectic eigenvalues and a Lidskii type theorem
Canadian journal of mathematics, Tome 74 (2022) no. 2, pp. 457-485

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DOI

Associated with every $2n\times 2n$ real positive definite matrix $A,$ there exist n positive numbers called the symplectic eigenvalues of $A,$ and a basis of $\mathbb {R}^{2n}$ called the symplectic eigenbasis of A corresponding to these numbers. In this paper, we discuss differentiability and analyticity of the symplectic eigenvalues and corresponding symplectic eigenbasis and compute their derivatives. We then derive an analogue of Lidskii’s theorem for symplectic eigenvalues as an application.
DOI : 10.4153/S0008414X2000084X
Mots-clés : Positive definite matrix, Williamson’s theorem, symplectic eigenvalue, symplectic eigenvector pair, derivative, analyticity, majorization, Lidskii’s theorem 
Jain, Tanvi; Mishra, Hemant Kumar. Derivatives of symplectic eigenvalues and a Lidskii type theorem. Canadian journal of mathematics, Tome 74 (2022) no. 2, pp. 457-485. doi: 10.4153/S0008414X2000084X
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     title = {Derivatives of symplectic eigenvalues and a {Lidskii} type theorem},
     journal = {Canadian journal of mathematics},
     pages = {457--485},
     year = {2022},
     volume = {74},
     number = {2},
     doi = {10.4153/S0008414X2000084X},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X2000084X/}
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