Frobenius' inequality

The inequality of Frobenius is a result of linear algebra , one of the branches of mathematics . It is named after Georg Frobenius and deals with the relationships between the ranks of three linear maps executed one after the other .

Formulation of the inequality

The inequality says the following:

Given are four vector spaces over an arbitrary body and three linear mappings    ,     and    . ${\displaystyle U,V,W,X}$ ${\displaystyle K}$ ${\displaystyle \alpha \colon U\to V}$ ${\displaystyle \beta \colon V\to W}$ ${\displaystyle \gamma \colon W\to X}$

Then:

${\displaystyle \mathrm {rg} (\gamma \beta )+\mathrm {rg} (\beta \alpha )\leq \mathrm {rg} (\beta )+\mathrm {rg} (\gamma \beta \alpha )}$  .

Evidence sketch

Let be a complementary space of   in   , so ${\displaystyle W_{0}}$ ${\displaystyle \mathrm {im} (\beta \alpha )}$ ${\displaystyle \mathrm {im} (\beta )}$

${\displaystyle \mathrm {im} (\beta )=\mathrm {im} (\beta \alpha )\oplus W_{0}}$ .

Then follows

${\displaystyle \mathrm {im} (\gamma \beta )=\mathrm {im} (\gamma \beta \alpha )+\gamma (W_{0})}$

and further

${\displaystyle \mathrm {rg} (\gamma \beta )\leq \mathrm {rg} (\gamma \beta \alpha )+\dim(\gamma (W_{0}))}$ .

So you get

${\displaystyle {\begin{aligned}\mathrm {rg} (\beta \alpha )+\mathrm {rg} (\gamma \beta )&\leq \mathrm {rg} (\beta \alpha )+\mathrm {rg} (\gamma \beta \alpha )+\dim(\gamma (W_{0}))\\&\leq \mathrm {rg} (\beta \alpha )+\mathrm {rg} (\gamma \beta \alpha )+\dim(W_{0})\\&=[\dim(\mathrm {im} (\beta \alpha ))+\dim(W_{0})]+\mathrm {rg} (\gamma \beta \alpha )\\&=\dim(\mathrm {im} (\beta ))+\mathrm {rg} (\gamma \beta \alpha )\\&=\mathrm {rg} (\beta )+\mathrm {rg} (\gamma \beta \alpha ),\end{aligned}}}$

so overall the claimed inequality.

annotation

Since the notion of dimension and the proof of the existence of a complementary space require the assumption of the validity of the axiom of choice for any vector spaces , in the event that one does not want to make this assumption, vector spaces of finite dimension must be assumed. For such, the inequality is always valid.

literature

• Hans-Joachim Kowalsky , Gerhard O. Michler : Lineare Algebra (=  De Gruyter textbook ). 12th, revised edition. Verlag Walter de Gruyter, Berlin (among others) 2003, ISBN 3-11-017963-6 .
• Günter Scheja , Uwe Storch : Textbook of Algebra. Including linear algebra. Part 1 (=  mathematical guidelines ). 2nd, revised and expanded edition. Teubner Verlag, Stuttgart 1994, ISBN 3-519-12203-0 ( MR1312830 ).

References and comments

1. ↑ Hans-Joachim Kowalsky , Gerhard O. Michler : Lineare Algebra (=  De Gruyter textbook ). 12th, revised edition. Verlag Walter de Gruyter, Berlin (among others) 2003, ISBN 3-11-017963-6 , p. 77-78, 375-376 .
2. ^ Günter Scheja , Uwe Storch : Textbook of Algebra. Including linear algebra. Part 1 (=  mathematical guidelines ). 2nd, revised and expanded edition. Teubner Verlag, Stuttgart 1994, ISBN 3-519-12203-0 , pp. 389 ( MR1312830 ).
3. ↑ For the sake of clarity of the formulas, instead of the representation of the composition in the form, the shorter multiplicative representation in the form and correspondingly in the other cases. ${\displaystyle \beta \circ \alpha }$ ${\displaystyle \beta \alpha }$