A spectral analysis is not required to find the rotation axis. If n denotes the unit eigenvector aligned with the rotation axis, and if α α denotes the rotation angle, then it can be shown that 2 sin ⁡ ⁡ ( α α ) n = { A 32 − − A 23 , A 13 − − A 31 , A 21 − − A 12 } . Consequently, the expense of an eigenvalue analysis can be avoided by simply normalizing this vector if it has a nonzero magnitude. On the other hand, if this vector has a zero magnitude, it means that sin ⁡ ⁡ ( α α ) = 0 . In other words, this vector will be zero if and only if the rotation angle is 0 or 180 degrees, and the rotation axis may be assigned in this case by normalizing any column of A + I that has a nonzero magnitude.