Bivector - Three Dimensions - Matrices

Matrices

Bivectors are isomorphic to skew-symmetric matrices; the general bivector B₂₃e₂₃ + B₃₁e₃₁ + B₁₂e₁₂ maps to the matrix

This multiplied by vectors on both sides gives the same vector as the product of a vector and bivector; an example is the angular velocity tensor.

Skew symmetric matrices generate orthogonal matrices with determinant 1 through the exponential map. In particular the exponent of a bivector associated with a rotation is a rotation matrix, that is the rotation matrix M_R given by the above skew-symmetric matrix is

The rotation described by M_R is the same as that described by the rotor R given by

and the matrix M_R can be also calculated directly from rotor R:

Bivectors are related to the eigenvalues of a rotation matrix. Given a rotation matrix M the eigenvalues can calculated by solving the characteristic equation for that matrix 0 = det (M - λI). By the fundamental theorem of algebra this has three roots, but only one real root as there is only one eigenvector, the axis of rotation. The other roots must be a complex conjugate pair. They have unit magnitude so purely imaginary logarithms, equal to the magnitude of the bivector associated with the rotation, which is also the angle of rotation. The eigenvectors associated with the complex eigenvalues are in the plane of the bivector, so the exterior product of two non-parallel eigenvectors result in the bivector, or at least a multiple of it.