Cartan Theorem - Dieudonne

Cartan – Dieudonne Theorem - a theorem named after the French mathematicians Eli Joseph Cartan and Jean Dieudonne . The theorem concerns the structure of automorphisms of a space equipped with a symmetric bilinear form (for example, Euclidean space ).

Statement of the theorem

Let ( V , b ) be an n- dimensional vector space (over a field whose characteristic is not equal to 2) with a non-degenerate symmetric bilinear form. Then each element of the orthogonal group O ( V , b ) is represented as a composition of no more than n symmetries with respect to hyperplanes.

Corollary of the theorem

If a $T$ ${\ displaystyle T}$ - orthogonal transformation to $R^{(3)}$ ${\ displaystyle R ^ {(3)}}$ and $|T|=1$ ${\ displaystyle | T | = 1}$ then there is a vector $\nu \neq 0$ ${\ displaystyle \ nu \ neq 0}$ such that $T_{\nu }=\nu$ ${\ displaystyle T _ {\ nu} = \ nu}$ .

Literature

Gallier JH Geometric Methods and Applications: For Computer Science and Engineering. - University of Pennsylvania: Springer Science + Business Media , 2001. - Vol. 38. - 565 p. - (Texts in applied mathematics). - ISBN 0387950443 .
Gallot S., Hulin D., Lafontaine J. Riemannian Geometry. - Springer Science + Business Media, 2004 .-- 322 p. - (Universitext Series). - ISBN 3540204938 .
Garling DJH Clifford Algebras: An Introduction. - Cambridge University Press , 2011. - Vol. 78. - 208 p. - (London Mathematical Society Student Texts). - ISBN 1107422191 .
Qit Yuan Lam . Introduction to Quadratic Forms Over Fields. - American Mathematical Society , 2005. - Vol. 67. - 550 p. - (Graduate studies in mathematics). - ISBN 0821810952 .

Cartan Theorem - Dieudonne

Statement of the theorem

Corollary of the theorem

Literature

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