as a check for simplicity. In particular the exterior product of bivectors only exists in four or more dimensions, so all bivectors in two and three dimensions are simple. General bivectors and Matrices Bivectors are isomorphic to skew-symmetric matrices; the general bivector maps to the matrix This multiplied by vectors on both sides gives the same vector as the product of a vector and bivector minus the outer product; 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 MR given by the above skew-symmetric matrix is The rotation described by MR is the same as that described by the rotor R given by and the matrix MR 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 . By the fundamental theorem of algebra this has three roots (only one of which is real as there is only one eigenvector, i.e., 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 results in the bivector (or a multiple thereof). Two dimensions When working with coordinates in geometric algebra it is usual to write the basis vectors as (e1, e2, ...), a convention that will be used here. A vector in real two-dimensional space R2 can be written , where a1 and a2 are real numbers, e1 and e2 are orthonormal basis vectors. The geometric product of two such vectors is This can be split into the symmetric, scalar-valued, scalar product and an antisymmetric, bivector-valued exterior product: All bivectors in two dimensions are of this form, that is multiples of the bivector e1e2, written e12 to emphasise it is a bivector rather than a vector. The magnitude of e12 is 1, with so it is called the unit bivector. The term unit bivector can be used in other dimensions but it is only uniquely