Graßmann-Plücker relation

The Graßmann-Plücker relations describe relationships between determinants with partially matching columns.

Definitions and sentences

General form

A general Graßmann-Plücker relation has the form

${\displaystyle \sum _{i=1}^{r+1}(-1)^{i}\cdot \det(A_{1},\dots ,A_{r-1},B_{i})\cdot \det(B_{1},\dots ,B_{i-1},B_{i+1},\dots ,B_{r+1})=0}$

where are vectors in an r - dimensional vector space that form the columns of the matrices whose determinants are calculated. ${\displaystyle A_{1},\dots ,A_{r-1},B_{1},\dots ,B_{r+1}}$

The dimension of the underlying vector space is often referred to as rank (and therefore abbreviated here as r ). In cases where the columns represent homogeneous coordinates of points, these points are one dimension lower in a projective space .

Concrete shape for low dimensions

In rank 2 the formula has 3 summands and uses 4 vectors A to D :

${\displaystyle \det(A,B)\cdot \det(C,D)-\det(A,C)\cdot \det(B,D)+\det(A,D)\cdot \det(B,C)=0}$

In rank 3 the formula has 4 summands and uses 6 vectors A to F :

${\displaystyle \det(A,B,C)\cdot \det(D,E,F)-\det(A,B,D)\cdot \det(C,E,F)+\det(A,B,E)\cdot \det(C,D,F)-\det(A,B,F)\cdot \det(C,D,E)=0}$

Brief, conceptual proof

We fixate and look at the picture ${\displaystyle A_{1},\dots ,A_{r-1}}$

${\displaystyle (B_{1},\dots ,B_{r+1})\mapsto {\text{ die linke Seite der Graßmann-Plücker-Relation}}}$

This mapping is apparently multilinear (i.e. linear in each separately). In addition, the right-hand side becomes zero if there is with , because then only the summands with and are possibly not equal to zero, and they cancel each other out. That is, the mapping is alternating. A fundamental theorem of linear algebra says that an alternating multilinear mapping of r + 1 vectors on an r -dimensional vector space must be identically zero. That is precisely the Graßmann-Plücker relation. ${\displaystyle B_{i}}$ ${\displaystyle j\neq k}$ ${\displaystyle B_{j}=B_{k}}$ ${\displaystyle i=j}$ ${\displaystyle i=k}$

Long proof

If all occurring summands are 0, the equation is trivially fulfilled. So let's assume that one of the summands is different from 0. OBdA is this the first summand, since we can rearrange the vectors of the two sets A and B at will. The first term therefore consists of two matrices whose determinants are different from 0.

Denoting the matrix in the first determinant of M and the second with N .

${\displaystyle M=(A_{1},\dots ,A_{r-1},B_{1})\quad N=(B_{2},\dots ,B_{r+1})}$

If you multiply all occurring matrices by the inverse matrix , each determinant is multiplied by the factor, i.e. the entire equation is multiplied by the square of it. This factor can be factored out and taken from the equation. Since the identity matrix is, one can o. B. d. A. Assume that the first matrix is ​​the identity matrix. ${\displaystyle M^{-1}}$ ${\displaystyle \det(M^{-1})\neq 0}$ ${\displaystyle M^{-1}M}$

In this case (for ) and . ${\displaystyle A_{i}=e_{i}}$ ${\displaystyle i=1,\dots ,(r-1)}$ ${\displaystyle B_{1}=e_{r}}$

${\displaystyle \sum _{i=1}^{r+1}(-1)^{i}\cdot \det(e_{1},\dots ,e_{r-1},B_{i})\cdot \det(B_{1},\dots ,B_{i-1},B_{i+1},\dots ,B_{r+1})=}$

${\displaystyle \det(E_{r})\cdot \det(B_{2},\dots ,B_{r+1})+\sum _{i=2}^{r+1}(-1)^{i}\cdot \det(e_{1},\dots ,e_{r-1},B_{i})\cdot \det(e_{r},\dots ,B_{i-1},B_{i+1},\dots ,B_{r+1})=}$

${\displaystyle \det(N)+\sum _{i=2}^{r+1}(-1)^{i}\cdot n_{r,(i-1)}\cdot \det(N_{r,(i-1)})=}$

${\displaystyle \det(N)-\det(N)=0}$

The sum is taken as the development of the determinant after the last line. The entry that is in the matrix in the last row and in the column corresponds to the last component of the vector , since it begins with . The matrix is the sub-matrix if you remove the vector and the last row. These sub-matrices result from the development of the second determinant after the first column. ${\displaystyle n_{r,(i-1)}}$ ${\displaystyle N}$ ${\displaystyle r}$ ${\displaystyle i}$ ${\displaystyle B_{i}}$ ${\displaystyle N}$ ${\displaystyle B_{2}}$ ${\displaystyle N_{r,(i-1)}}$ ${\displaystyle B_{i}}$

Applications

• The Graßmann-Plücker relations are among the syzygies . They can be used to formulate proofs (e.g. of geometric inference theorems).
• Oriented matroids can be characterized by the fact that they are in no obvious contradiction to the Graßmann-Plücker relations.
• Graßmann-Plücker coordinates , which are used to describe geometric objects in higher-dimensional projective spaces , have to fulfill these relations in order to be consistent.

See also

• Syzygy

literature

• Hermann Graßmann : The linear expansion theory . 1st edition. 1844 ( PDF, 11MB ).
• Jürgen Richter-Gebert and Thorsten Orendt: Geometry calculi . 1st edition. Springer, Berlin Heidelberg 2009, ISBN 978-3-642-02529-7 .

Individual evidence

1. ^ Geometry calculi , pp. 141 ff.
2. ^ Geometry calculi , p. 142 f.