pair is associated with a simple component of the bivector associated with the rotation. In particular the log of each pair is ± the magnitude, while eigenvectors generated from the roots are parallel to and so can be used to generate the bivector. In general the eigenvalues and bivectors are unique, and the set of eigenvalues gives the full decomposition into simple bivectors; if roots are repeated then the decomposition of the bivector into simple bivectors is not unique. Projective geometry Geometric algebra can be applied to projective geometry in a straightforward way. The geometric algebra used is , the algebra of the real vector space Rn. This is used to describe objects in the real projective space RPn−1. The non-zero vectors in Cln(R) or Rn are associated with points in the projective space so vectors that differ only by a scale factor, so their exterior product is zero, map to the same point. Non-zero simple bivectors in ⋀2Rn represent lines in RPn−1, with bivectors differing only by a (positive or negative) scale factor representing the same line. A description of the projective geometry can be constructed in the geometric algebra using basic operations. For example, given two distinct points in RPn−1 represented by vectors a and b the line containing them is given by (or ). Two lines intersect in a point if for their bivectors A and B. This point is given by the vector The operation "∨" is the meet, which can be defined as above in terms of the join, for non-zero . Using these operations projective geometry can be formulated in terms of geometric algebra. For example, given a third (non-zero) bivector C the point p lies on the line given by C if and only if So the condition for the lines given by A, B and C to be collinear is which in Cl3(R) and RP2 simplifies to where the angle brackets denote the scalar part of the geometric product. In the same way all projective space operations can be written in terms of geometric algebra, with bivectors representing general lines in projective space, so the whole geometry can be developed using geometric algebra. Tensors and matrices As noted above a bivector can be written as a skew-symmetric matrix, which through the exponential map generates a rotation matrix that describes the same rotation as the rotor, also generated by the exponential map but