# American Institute of Mathematical Sciences

2016, 6(2): 115-126. doi: 10.3934/naco.2016003

## Derivatives of eigenvalues and Jordan frames

 1 Departamento de Matemática, Faculdade de Ciências e Tecnologia & CMA, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal

Received  January 2015 Revised  May 2016 Published  June 2016

Every element in a Euclidean Jordan algebra has a spectral decomposition. This spectral decomposition is generalization of the spectral decompositions of a matrix. In the context of Euclidean Jordan algebras, this is written using eigenvalues and the so-called Jordan frame. In this paper we deduce the derivative of eigenvalues in the context of Euclidean Jordan algebras. We also deduce the derivative of the elements of a Jordan frame associated to the spectral decomposition.
Citation: Manuel V. C. Vieira. Derivatives of eigenvalues and Jordan frames. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 115-126. doi: 10.3934/naco.2016003
##### References:

show all references

##### References:
 [1] Yu-Lin Chang, Chin-Yu Yang. Some useful inequalities via trace function method in Euclidean Jordan algebras. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 39-48. doi: 10.3934/naco.2014.4.39 [2] Golamreza Zamani Eskandani, Hamid Vaezi. Hyers--Ulam--Rassias stability of derivations in proper Jordan $CQ^{*}$-algebras. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1469-1477. doi: 10.3934/dcds.2011.31.1469 [3] M. P. de Oliveira. On 3-graded Lie algebras, Jordan pairs and the canonical kernel function. Electronic Research Announcements, 2003, 9: 142-151. [4] Xiao-Hong Liu, Wei Wu. Coerciveness of some merit functions over symmetric cones. Journal of Industrial & Management Optimization, 2009, 5 (3) : 603-613. doi: 10.3934/jimo.2009.5.603 [5] Vincent Astier, Thomas Unger. Signatures, sums of hermitian squares and positive cones on algebras with involution. Electronic Research Announcements, 2018, 25: 16-26. doi: 10.3934/era.2018.25.003 [6] Lixing Han. An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 583-599. doi: 10.3934/naco.2013.3.583 [7] Xiao-Hong Liu, Wei-Zhe Gu. Smoothing Newton algorithm based on a regularized one-parametric class of smoothing functions for generalized complementarity problems over symmetric cones. Journal of Industrial & Management Optimization, 2010, 6 (2) : 363-380. doi: 10.3934/jimo.2010.6.363 [8] Brigitte Vallée. Euclidean dynamics. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 281-352. doi: 10.3934/dcds.2006.15.281 [9] Thiago Ferraiol, Mauro Patrão, Lucas Seco. Jordan decomposition and dynamics on flag manifolds. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 923-947. doi: 10.3934/dcds.2010.26.923 [10] Víctor Ayala, Adriano Da Silva, Philippe Jouan. Jordan decomposition and the recurrent set of flows of automorphisms. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1543-1559. doi: 10.3934/dcds.2020330 [11] A. A. Kirillov. Family algebras. Electronic Research Announcements, 2000, 6: 7-20. [12] Martin Brokate, Pavel Krejčí. Weak differentiability of scalar hysteresis operators. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2405-2421. doi: 10.3934/dcds.2015.35.2405 [13] Misha Guysinsky, Boris Hasselblatt, Victoria Rayskin. Differentiability of the Hartman--Grobman linearization. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 979-984. doi: 10.3934/dcds.2003.9.979 [14] Frédéric Naud. Birkhoff cones, symbolic dynamics and spectrum of transfer operators. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 581-598. doi: 10.3934/dcds.2004.11.581 [15] Pooja Louhan, S. K. Suneja. On fractional vector optimization over cones with support functions. Journal of Industrial & Management Optimization, 2017, 13 (2) : 549-572. doi: 10.3934/jimo.2016031 [16] Songting Luo, Leonidas J. Guibas, Hong-Kai Zhao. Euclidean skeletons using closest points. Inverse Problems & Imaging, 2011, 5 (1) : 95-113. doi: 10.3934/ipi.2011.5.95 [17] A. S. Dzhumadil'daev. Jordan elements and Left-Center of a Free Leibniz algebra. Electronic Research Announcements, 2011, 18: 31-49. doi: 10.3934/era.2011.18.31 [18] Steffen Konig and Changchang Xi. Cellular algebras and quasi-hereditary algebras: a comparison. Electronic Research Announcements, 1999, 5: 71-75. [19] Alexandre Rocha, Mário Jorge Dias Carneiro. A dynamical condition for differentiability of Mather's average action. Journal of Geometric Mechanics, 2014, 6 (4) : 549-566. doi: 10.3934/jgm.2014.6.549 [20] Xia Li, Yong Wang, Zheng-Hai Huang. Continuity, differentiability and semismoothness of generalized tensor functions. Journal of Industrial & Management Optimization, 2021, 17 (6) : 3525-3550. doi: 10.3934/jimo.2020131

Impact Factor: