 Exceptionally, a rotation block may be diagonal, .Thus, negating one column if necessary, and noting that a reflection diagonalizes to a +1 and −1, any orthogonal matrix can be brought to the form

The matrices give conjugate pairs of eigenvalues lying on the unit circle in the complex plane; so this decomposition confirms that all eigenvalues have absolute value 1.If is odd, there is at least one real eigenvalue, +1 or −1; for a rotation, the eigenvector associated with +1 is the rotation axis.