KreuzMatrix

KreuzMatrix is a mathematical construct used to represent the cross product operation in matrix form. Given a three-dimensional vector v = (v1, v2, v3), the KreuzMatrix [v]× is the 3×3 skew-symmetric matrix defined by

[v]× = [ 0 -v3 v2

v3 0 -v1

-v2 v1 0 ].

For any vector w, [v]× w = v × w, so the cross product can be expressed as a simple matrix multiplication.

Properties of the KreuzMatrix include linear dependence on v, antisymmetry ([v]×^T = -[v]×), and the ability to encode cross products in a linear-algebraic framework. For nonzero v, det([v]×) = 0 and rank([v]×) = 2. The eigenvalues are 0 and a pair of purely imaginary eigenvalues ±i‖v‖. The matrix is a standard example of a skew-symmetric matrix and is closely connected to the Lie algebra so(3).

Applications of the KreuzMatrix appear across several fields. It provides a compact algebraic tool in rigid body dynamics, robotics, and computer graphics, where cross products frequently arise. In the context of Lie theory, [v]× represents an infinitesimal rotation in so(3), and the matrix exponential exp([v]×θ) yields a finite rotation about the axis v by angle θ (Rodrigues' formula). The concept also underpins algorithms for orientation, angular velocity, and rotational kinematics.

See also: cross product, skew-symmetric matrix, so(3). The term KreuzMatrix reflects the role of the cross product in its construction.