# 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 and Optimization, 2016, 6 (2) : 115-126. doi: 10.3934/naco.2016003
##### References:
 [1] M. Baes, Convexity and differentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras, Linear Algebra Appl., 422 (2007), 664-700. doi: 10.1016/j.laa.2006.11.025. [2] Y. Q. Bai, M. El Ghami and C. Roos, A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization, SIAM Journal on Optimization, 15 (2004), 101-128. doi: 10.1137/S1052623403423114. [3] J. Faraut and A. Korányi, Analysis on Symmetric Cones, The Clarendon Press Oxford University Press, New York, 1994. [4] L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithms, J. Comput. Appl. Math., 86 (1997), 149-175. doi: 10.1016/S0377-0427(97)00153-2. [5] L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, Positivity, 1 (1997), 331-357. doi: 10.1023/A:1009701824047. [6] L. Faybusovich and R. Arana, A long-step primal-dual algorithm for the symmetric programming problem, Systems Control Lett., 43 (2001), 3-7. doi: 10.1016/S0167-6911(01)00092-5. [7] L. Faybusovich, A Jordan-algebraic approach to potential-reduction algorithms, Math. Z., 239 (2002), 117-129. doi: 10.1007/s002090100286. [8] G. Gu, M. Zangiabadi and C. Roos, Full Nesterov-Todd step infeasible interior-point method for symmetric optimization, European Journal of Operational Research, 214 (2011), 473-484. doi: 10.1016/j.ejor.2011.02.022. [9] R. A. Hauser and O. Güler, Self-scaled barrier functions on symmetric cones and their classification, Found. Comput. Math., 2 (2002), 121-143. [10] R. A. Hauser and Y. Lim, Self-scaled barriers for irreducible symmetric cones, SIAM J. Optim., 12 (2002), 715-723. doi: 10.1137/S1052623400370953. [11] Z.-H. Huang and T. Ni, Smoothing algorithms for complementarity problems over symmetric cones, Comput. Optim. Appl., 450 (2010), 557-579. doi: 10.1007/s10589-008-9180-y. [12] B. Kheirfam, A corrector-predictor path-following method for convex quadratic symmetric cone optimization, Journal of Optimization Theory and Applications, 164 (2015), 246-260. doi: 10.1007/s10957-014-0554-2. [13] M. Kojima and M. Muramatsu, An extension of sums of squares relaxations to polynomial optimization problems over symmetric cones, Math. Program., 110 (2007), 315-336. doi: 10.1007/s10107-006-0004-5. [14] P. Lancaster, On eigenvalues of matrices dependent on a parameter, Numer. Math., 6 (1964), 377-387. [15] G. Lesaja and C. Roos, Kernel-based interior-point methods for monotone linear complementarity problems over symmetric cones, Journal of Optimization Theory and Applications, 150 (2011), 444-474. doi: 10.1007/s10957-011-9848-9. [16] H. Liu, X. Yang and C. Liu, A new wide neighborhood primal-dual infeasible-interior-point method for symmetric cone programming, Journal of Optimization Theory and Applications, 158 (2013), 796-815. doi: 10.1007/s10957-013-0303-y. [17] Y. Nesterov and A. Nemirovskii, Interior Point Polynomial Algorithms in Convex Programming, SIAM: Studies in Applied and Numerical Mathematics, Philadelphia, 1994. doi: 10.1137/1.9781611970791. [18] S. H. Schmieta and F. Alizadeh, Extension of primal-dual interior point algorithms to symmetric cones, Math. Program., 96 (2003), 409-438. doi: 10.1007/s10107-003-0380-z. [19] S. H. Schmieta and F. Alizadeh, Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones, Math. Oper. Res., 26 (2001), 543-564. doi: 10.1287/moor.26.3.543.10582. [20] M. V. C. Vieira, Interior-point methods based on kernel functions for symmetric optimization, Optimization Methods and Software, 27 (2012), 513-537. doi: 10.1080/10556788.2010.544877. [21] G. Q. Wang, C. J. Yub and K. L. Teo, A new full Nesterov-Todd step feasible interior-point method for convex quadratic symmetric cone optimization, Applied Mathematics and Computation, 221 (2013), 329-343. doi: 10.1016/j.amc.2013.06.064. [22] Z. Yu, Y. Zhu and Q. Cao, On the convergence of central path and generalized proximal point method for symmetric cone linear programming, Applied Mathematics & Information Sciences, 7 (2103), 2327-2333. doi: 10.12785/amis/070624.

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##### References:
 [1] M. Baes, Convexity and differentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras, Linear Algebra Appl., 422 (2007), 664-700. doi: 10.1016/j.laa.2006.11.025. [2] Y. Q. Bai, M. El Ghami and C. Roos, A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization, SIAM Journal on Optimization, 15 (2004), 101-128. doi: 10.1137/S1052623403423114. [3] J. Faraut and A. Korányi, Analysis on Symmetric Cones, The Clarendon Press Oxford University Press, New York, 1994. [4] L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithms, J. Comput. Appl. Math., 86 (1997), 149-175. doi: 10.1016/S0377-0427(97)00153-2. [5] L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, Positivity, 1 (1997), 331-357. doi: 10.1023/A:1009701824047. [6] L. Faybusovich and R. Arana, A long-step primal-dual algorithm for the symmetric programming problem, Systems Control Lett., 43 (2001), 3-7. doi: 10.1016/S0167-6911(01)00092-5. [7] L. Faybusovich, A Jordan-algebraic approach to potential-reduction algorithms, Math. Z., 239 (2002), 117-129. doi: 10.1007/s002090100286. [8] G. Gu, M. Zangiabadi and C. Roos, Full Nesterov-Todd step infeasible interior-point method for symmetric optimization, European Journal of Operational Research, 214 (2011), 473-484. doi: 10.1016/j.ejor.2011.02.022. [9] R. A. Hauser and O. Güler, Self-scaled barrier functions on symmetric cones and their classification, Found. Comput. Math., 2 (2002), 121-143. [10] R. A. Hauser and Y. Lim, Self-scaled barriers for irreducible symmetric cones, SIAM J. Optim., 12 (2002), 715-723. doi: 10.1137/S1052623400370953. [11] Z.-H. Huang and T. Ni, Smoothing algorithms for complementarity problems over symmetric cones, Comput. Optim. Appl., 450 (2010), 557-579. doi: 10.1007/s10589-008-9180-y. [12] B. Kheirfam, A corrector-predictor path-following method for convex quadratic symmetric cone optimization, Journal of Optimization Theory and Applications, 164 (2015), 246-260. doi: 10.1007/s10957-014-0554-2. [13] M. Kojima and M. Muramatsu, An extension of sums of squares relaxations to polynomial optimization problems over symmetric cones, Math. Program., 110 (2007), 315-336. doi: 10.1007/s10107-006-0004-5. [14] P. Lancaster, On eigenvalues of matrices dependent on a parameter, Numer. Math., 6 (1964), 377-387. [15] G. Lesaja and C. Roos, Kernel-based interior-point methods for monotone linear complementarity problems over symmetric cones, Journal of Optimization Theory and Applications, 150 (2011), 444-474. doi: 10.1007/s10957-011-9848-9. [16] H. Liu, X. Yang and C. Liu, A new wide neighborhood primal-dual infeasible-interior-point method for symmetric cone programming, Journal of Optimization Theory and Applications, 158 (2013), 796-815. doi: 10.1007/s10957-013-0303-y. [17] Y. Nesterov and A. Nemirovskii, Interior Point Polynomial Algorithms in Convex Programming, SIAM: Studies in Applied and Numerical Mathematics, Philadelphia, 1994. doi: 10.1137/1.9781611970791. [18] S. H. Schmieta and F. Alizadeh, Extension of primal-dual interior point algorithms to symmetric cones, Math. Program., 96 (2003), 409-438. doi: 10.1007/s10107-003-0380-z. [19] S. H. Schmieta and F. Alizadeh, Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones, Math. Oper. Res., 26 (2001), 543-564. doi: 10.1287/moor.26.3.543.10582. [20] M. V. C. Vieira, Interior-point methods based on kernel functions for symmetric optimization, Optimization Methods and Software, 27 (2012), 513-537. doi: 10.1080/10556788.2010.544877. [21] G. Q. Wang, C. J. Yub and K. L. Teo, A new full Nesterov-Todd step feasible interior-point method for convex quadratic symmetric cone optimization, Applied Mathematics and Computation, 221 (2013), 329-343. doi: 10.1016/j.amc.2013.06.064. [22] Z. Yu, Y. Zhu and Q. Cao, On the convergence of central path and generalized proximal point method for symmetric cone linear programming, Applied Mathematics & Information Sciences, 7 (2103), 2327-2333. doi: 10.12785/amis/070624.
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