Article PDF
Avoid common mistakes on your manuscript.
References
Cf.T. Levi-Civita andU. Amaldi,Lezioni di meccanica razionale, Vol. 2, Part II (1927), pp. 308–311.
Cf.T. Levi-Civita andU. Amaldi, loc. cit., pp. 310–316 and 324–327.
Cf.G. Frobenius, Ueber die principale Transformation der Thetafunktionen mehrerer Variablen, « Journal für reine und angewandte Mathematik », Vol. 95 (1883), pp. 264–296, § 1, etc. orC. Jordan,Traité des substitutions, 1870, Chap. II, § 8.
G. Frobenius, Ueber die schiefe Invariante einer bilinearen oder quadratischen Form, « Journal für reine und angewandte Mathematik », Vol. 86 (1876), pp. 44–71, more particularly p. 48.
Cf. SirW. Thomson andP. G. Tait,Treatise on Natural Philosophy, Vol. 1, Part I (1879), pp. 389–396.
K. Weierstrass,Mathematische Werke, Vol. 1 (1894), pp. 233–256.
Cf.T. Levi-Civita andU. Amaldi, loc. cit., p. 333.
L. Autonne, Sur l' Hermitien, « Rendiconti del Circolo Matematico di Palermo », Vol. 16 (1902), pp. 104–128, more particularly pp. 123–125. The considerations ofAutonne concern the complex domain but are valid in the real domain also. The uniqueness of the polar factorization is not pointed out byAutonne. He proves, however, (loc. cit., pp. 120–121) that there exists but one positive definite matrixP=P′ such thatAA′=P 2. NowAA′=P 2 is a consequence ofA=PR. HenceP is completely determined byA. ConsequentlyR=P −1 A also is unique. This situation has been pointed out by the present author,On Non-Singular Bounded Matrices, « American Journal of Mathematics », Vol. 54 (1932), pp. 145–149.
G. D. Birkhoff,Dynamical Systems, 1927, Chap. III, p. 89.
Cf., e. g.,H. Weyl,The Theory of Groups and Quantum Mechanics (1931), Appendix.
Cf., e. g.,L. P. Eisenhart,Continuous Groups of Transformations (1933), Chap. I.
Cf.L. Autonne, loc. cit., pp. 120–121. The symmetric matrix σ having a square=v may be chosen as a positive definite matrix. This is, however, not needed in the above proof.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wintner, A. On the linear conservative dynamical systems. Annali di Matematica 13, 105–112 (1934). https://doi.org/10.1007/BF02413437
Issue Date:
DOI: https://doi.org/10.1007/BF02413437