Flag set

The flag theorem or trigonalization theorem is a doctrine of linear algebra , one of the sub-areas of mathematics . It arises in connection with the treatment of the so-called normal form problem , in which the possibility of normal form representations of Vektorraumendomorphismen by special matrices is investigated. The theorems on Jordan normal form also belong to this group of topics .

Formulation of the sentence

The theorem can be formulated as follows:

For a vector space endomorphism on a finite-dimensional vector space , the following conditions are equivalent: ${\displaystyle \phi \colon V\to V}$ ${\displaystyle V}$

(i) to exist in a flag   ( ) which -stable is in the sense that each of the occurring in this flag subspaces of in itself displayed is: ${\displaystyle \phi }$ ${\displaystyle V}$ ${\displaystyle \{0\}=V_{0}\subset V_{1}\subset \ldots \subset V_{n}=V}$ ${\displaystyle n=dim(V)}$ ${\displaystyle \phi }$ ${\displaystyle \phi }$

${\displaystyle \phi (V_{i})\subseteq V_{i}}$   ( ) ${\displaystyle i=0,\ldots ,n}$

(ii) The characteristic polynomial of decomposes into linear factors . ${\displaystyle \chi _{\varphi }}$ ${\displaystyle \phi }$

(iii) The minimal polynomial of decomposes into linear factors . ${\displaystyle \mu _{\varphi }}$ ${\displaystyle \phi }$

(iv) is trigonalizable . ${\displaystyle \phi }$

Inference

From the standard theorem (and taking into account the fundamental theorem of algebra ) the following corollary results :

In a finite-dimensional vector space over an algebraically closed field (and especially over the field of complex numbers !) Every endomorphism can be trigonalized.

Related sentence

The following result is closely related to the flag theorem , which gives a criterion for the diagonalisability of an endomorphism of the finite-dimensional vector space and states the following: ${\displaystyle \phi \colon V\to V}$ ${\displaystyle V}$

${\displaystyle \phi }$ can be diagonalized if and only if the minimal polynomial breaks down into linear factors, all of which are simple . ${\displaystyle \mu _{\varphi }}$

literature

• Gerd Fischer : Linear Algebra . 17th updated edition. Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-0996-4 .
• E. Lamprecht : Lineare Algebra I (=  Uni-Taschenbücher . Volume 1021 ). Uni-Taschenbücher, Basel (et al.) 1980, ISBN 3-7643-1175-4 .
• Eberhard Oeljeklaus, Reinhold Remmert : Lineare Algebra I (=  Heidelberg pocket books . Volume 150 ). Springer Verlag, Berlin ( inter alia ) 1974, ISBN 3-540-06715-9 ( MR0366944 ).
• Uwe Storch , Hartmut Wiebe: Textbook of Mathematics (=  spectrum textbook . Volume 2 : Linear Algebra ). 2nd, corrected edition. Spectrum Academic Publishing House, Heidelberg / Berlin 2010, ISBN 978-3-8274-2667-3 .
• Guido Walz [Red.]: Lexicon of Mathematics in six volumes . tape 5 . Spectrum Academic Publishing House, Heidelberg / Berlin 2002, ISBN 3-8274-0437-1 .

References and comments

1. ↑ Oeljeklaus-Remmert: p. 241 ff.
2. ^ Lamprecht: p. 139
3. ↑ ^{a b} Fischer: p. 242 ff.
4. ↑ Storch-Wiebe: p. 317
5. ^ Lexicon of Mathematics in six volumes. Fifth volume . S. 241 .
6. ↑ Instead of -stable , such a flag is also called -invariant . ${\displaystyle \phi }$ ${\displaystyle \phi }$
7. ↑ Such a linear operator is called instead trigonalisierbar also triangulated ; see. Lexicon of Mathematics , Vol. 5, p. 241.
8. ↑ Storch-Wiebe: p. 318
9. ↑ Storch-Wiebe: pp. 315-316