Skew-symmetric 4 × 4 matrix, which characterizes a straight line in projective space
The Plücker matrix is a special skew-symmetric 4 × 4 matrix, which characterizes a straight line in projective space. The matrix is defined by 6 Plücker coordinates with 4 degrees of freedom. It is named after the German mathematician Julius Plücker.
Definition.
A straight line in space is defined by two distinct points formula_1 and formula_2 in homogeneous coordinates of the projective space. Its Plücker matrix is:
formula_3
Where the skew-symmetric formula_4-matrix is defined by the 6 Plücker coordinates 
formula_5
with
formula_6
Plücker coordinates fulfill the Grassmann–Plücker relations
formula_7
and are defined up to scale. A Plücker matrix has only rank 2 and four degrees of freedom (just like lines in formula_8). They are independent of a particular choice of the points formula_9 and formula_10 and can be seen as a generalization of the line equation i.e. of the cross product for both the intersection (meet) of two lines, as well as the joining line of two points in the projective plane.
Properties.
The Plücker matrix allows us to express the following geometric operations as matrix-vector product:
Uniqueness.
Two arbitrary distinct points on the line can be written as a linear combination of formula_9 and formula_10:
formula_24
Their Plücker matrix is thus:
formula_25
up to scale identical to formula_26.
Intersection with a plane.
Let formula_27 denote the plane with the equation
formula_28
which does not contain the line formula_13. Then, the matrix-vector product with the Plücker matrix describes a point
formula_30
which lies on the line formula_13 because it is a linear combination of formula_9 and formula_10. formula_18 is also contained in the plane formula_14
formula_36
and must therefore be their point of intersection.
In addition, the product of the Plücker matrix with a plane is the zero-vector, exactly if the line formula_13 is contained entirely in the plane:
formula_38 contains formula_39
Dual Plücker matrix.
In projective three-space, both points and planes have the same representation as 4-vectors and the algebraic description of their geometric relationship (point lies on plane) is symmetric. By interchanging the terms plane and point in a theorem, one obtains a dual theorem which is also true.
In case of the Plücker matrix, there exists a dual representation of the line in space as the intersection of two planes:
formula_40
and
formula_41
in homogeneous coordinates of projective space. Their Plücker matrix is:
formula_42
and
formula_43
describes the plane formula_44 which contains both the point formula_18 and the line formula_13.
Relationship between primal and dual Plücker matrices.
As the vector formula_12, with an arbitrary plane formula_14, is either the zero-vector or a point on the line, it follows:
formula_49
Thus:
formula_50
The following product fulfills these properties:
formula_51
due to the Grassmann–Plücker relation. With the uniqueness of Plücker matrices up to scalar multiples, for the primal Plücker coordinates
formula_52
we obtain the following dual Plücker coordinates:
formula_53
In the projective plane.
The 'join' of two points in the projective plane is the operation of connecting two points with a straight line. Its line equation can be computed using the cross product:
formula_54
Dually, one can express the 'meet', or intersection of two straight lines by the cross-product:
formula_55
The relationship to Plücker matrices becomes evident, if one writes the cross product as a matrix-vector product with a skew-symmetric matrix:
formula_56
and analogously formula_57
Geometric interpretation.
Let formula_58 and formula_59, then we can write
formula_60
and
formula_61
where formula_62 is the displacement and formula_63 is the moment of the line, compare the geometric intuition of Plücker coordinates.