Convex SO ( N ) × SO ( n ) -invariant functions and refinements of von Neumann’s inequality

Bernard Dacorogna[1]; Pierre Maréchal[2]

  • [1] EPFL, CH-1015 Lausanne, Switzerland
  • [2] Université Paul Sabatier, Institut de mathématiques, F-31062 Toulouse cedex 9, France

Annales de la faculté des sciences de Toulouse Mathématiques (2007)

  • Volume: 16, Issue: 1, page 71-89
  • ISSN: 0240-2963

Abstract

top
A function  f on M N × n ( ℝ ) which is SO ( N ) × SO ( n ) -invariant is convex if and only if its restriction to the subspace of diagonal matrices is convex. This results from Von Neumann type inequalities and appeals, in the case where N = n , to the notion of signed singular value.

How to cite

top

Dacorogna, Bernard, and Maréchal, Pierre. "Convex $\operatorname{SO}(N)\times \operatorname{SO}(n)$-invariant functions and refinements of von Neumann’s inequality." Annales de la faculté des sciences de Toulouse Mathématiques 16.1 (2007): 71-89. <http://eudml.org/doc/10038>.

@article{Dacorogna2007,
abstract = {A function $f$ on $M_\{N\times n\}(\{\mathbb\{R\}\})$ which is $\mathop \{\mathrm\{SO\}(N)\}\times \mathop \{\mathrm\{SO\}(n)\}$-invariant is convex if and only if its restriction to the subspace of diagonal matrices is convex. This results from Von Neumann type inequalities and appeals, in the case where $N=n$, to the notion of signed singular value.},
affiliation = {EPFL, CH-1015 Lausanne, Switzerland; Université Paul Sabatier, Institut de mathématiques, F-31062 Toulouse cedex 9, France},
author = {Dacorogna, Bernard, Maréchal, Pierre},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {singular values; von Neumann's inequality; convex -invariant functions},
language = {eng},
number = {1},
pages = {71-89},
publisher = {Université Paul Sabatier, Toulouse},
title = {Convex $\operatorname\{SO\}(N)\times \operatorname\{SO\}(n)$-invariant functions and refinements of von Neumann’s inequality},
url = {http://eudml.org/doc/10038},
volume = {16},
year = {2007},
}

TY - JOUR
AU - Dacorogna, Bernard
AU - Maréchal, Pierre
TI - Convex $\operatorname{SO}(N)\times \operatorname{SO}(n)$-invariant functions and refinements of von Neumann’s inequality
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2007
PB - Université Paul Sabatier, Toulouse
VL - 16
IS - 1
SP - 71
EP - 89
AB - A function $f$ on $M_{N\times n}({\mathbb{R}})$ which is $\mathop {\mathrm{SO}(N)}\times \mathop {\mathrm{SO}(n)}$-invariant is convex if and only if its restriction to the subspace of diagonal matrices is convex. This results from Von Neumann type inequalities and appeals, in the case where $N=n$, to the notion of signed singular value.
LA - eng
KW - singular values; von Neumann's inequality; convex -invariant functions
UR - http://eudml.org/doc/10038
ER -

References

top
  1. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Archives For Rational Mechanics and Analysis, 63, p. 337-403 (1977). Zbl0368.73040MR475169
  2. B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, 1989. Zbl0703.49001MR990890
  3. B. Dacorogna, P. Marcellini, Implicit Partial Differential Equations, Birkhäuser, 1999. Zbl0938.35002MR1702252
  4. B. Dacorogna, H. Koshigoe, On the different notions of convexity for rotationally invariant functions, Annales de la Faculté des Sciences de Toulouse, II(2), p. 163-184 (1993). Zbl0828.49016MR1253387
  5. J.-B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Minimization Algorithms, I and II, Springer-Verlag, 1993. Zbl0795.49002MR1261420
  6. R. A. Horn, C. A. Johnson, Matrix Analysis, Cambridge University Press, 1985. Zbl0576.15001MR832183
  7. B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Annales Scientifiques de l’Ecole Normale Supérieure, 6, p. 413-455 (1973). Zbl0293.22019MR364552
  8. P. J. Laurent, Approximation et Optimisation, Hermann, 1972. Zbl0238.90058MR467080
  9. H. Le Dret, Sur les fonctions de matrices convexes et isotropes, Comptes Rendus de l’Académie des Sciences, Paris, Série 1, Mathématiques, 310, p. 617-620 (1990). Zbl0693.15013MR1050144
  10. A. Lewis, Group invariance and convex matrix analysis, SIAM Journal of Matrix Analysis and Applications, 17, p. 927-949 (1996). Zbl0876.15016MR1410709
  11. A. Lewis, Convex analysis on Cartan subspaces, Nonlinear Analysis, 42, p. 813-820 (2000). Zbl1159.15303MR1776925
  12. A. Lewis, The mathematics of eigenvalue optimization, Mathematical Programming, Series B 97, p. 155-176 (2003). Zbl1035.90085MR2004395
  13. P. Rosakis, Characterization of convex isotropic functions, Journal of Elasticity, 49, p. 257-267 (1997). Zbl0906.73018MR1633494
  14. R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970. Zbl0193.18401MR274683
  15. A. Seeger, Convex analysis of spectrally defined matrix functions, SIAM Journal on Optimization, 7(3), p. 679-696 (1997). Zbl0890.15018MR1462061
  16. D. Serre, Matrices: Theory and Applications, Grad. Text in Math. 216, Springer-Verlag, 2002. See also http://www.umpa.ens-lyon.fr/ ∼ serre/publi.html. Zbl1011.15001MR1923507
  17. F. Vincent, Une note sur les fonctions convexes invariantes, Annales de la Faculté des Sciences de Toulouse, p. 357-363 (1997). Zbl0915.17007MR1611773

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.