This discussion applies to a proper rotation, and hence det A = 1 . Any improper orthogonal 3x3 matrix B may be written as B = − − A , in which A is proper orthogonal. That is, any improper orthogonal 3x3 matrix may be decomposed as a proper rotation (from which an axis of rotation can be found as described above) followed by an inversion (multiplication by −1). It follows that the rotation axis of A is also the eigenvector of B corresponding to an eigenvalue of −1.