the plane by negative φ radians. Inverse Let Since where the equivalence is in the projective linear group over quaternions, the inverse of f(u, 1) is Since homographies are bijections, maps the vector quaternions to the 3-sphere of versors. As versors represent rotations in 3-space, the homography f −1 produces rotations from the ball in ℝ3. Matrix map Among n×n square matrices over the reals, with I the identity matrix, let A be any skew-symmetric matrix (so that AT = −A). Then I + A is invertible, and the Cayley transform produces an orthogonal matrix, Q (so that QTQ = I). The matrix multiplication in the definition of Q above is commutative, so Q can be alternatively defined as . In fact, Q must have determinant +1, so is special orthogonal. Conversely, let Q be any orthogonal matrix which does not have −1 as an eigenvalue; then is a skew-symmetric matrix. The condition on Q automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices. A slightly different form is also seen, requiring different mappings in each direction, The mappings may also be written with the order of the factors reversed; however, A always commutes with (μI ± A)−1, so the reordering does not affect the definition. Examples In the 2×2 case, we have The 180° rotation matrix, −I, is excluded, though it is the limit as tan θ⁄2 goes to infinity. In the 3×3 case, we have where K = w2 + x2 + y2 + z2, and where w = 1. This we recognize as the rotation matrix corresponding to quaternion (by a formula Cayley had published the year before), except scaled so that w = 1 instead of the usual scaling so that w2 + x2 + y2 + z2 = 1. Thus vector (x,y,z) is the unit axis of rotation scaled by tan θ⁄2. Again excluded are 180° rotations, which in this case are all Q which are symmetric (so that QT = Q). Other matrices One can extend the mapping to complex matrices by substituting "unitary" for "orthogonal" and "skew-Hermitian" for "skew-symmetric", the difference being that the transpose (·T) is replaced by the conjugate transpose (·H). This is consistent with replacing the standard real inner product with the standard complex inner product. In fact, one may extend the definition further with choices of adjoint other than transpose or conjugate transpose. Formally, the