stretching it. If we write A in this basis, it is diagonal; but a diagonal orthogonal matrix is made of just +1s and −1s in the diagonal entries. Therefore, we don't have a proper rotation, but either the identity or the result of a sequence of reflections. It follows, then, that a proper rotation has some complex eigenvalue. Let v be the corresponding eigenvector. Then, as we showed in the previous topic, is also an eigenvector, and and are such that their scalar product vanishes: because, since is real, it equals its complex conjugate , and and are both representations of the same scalar product between and . This means and are orthogonal vectors. Also, they are both real vectors by construction. These vectors span the same subspace as and , which is an invariant subspace under the application of A. Therefore, they span an invariant plane. This plane is orthogonal to the invariant axis, which corresponds to the remaining eigenvector of A, with eigenvalue 1, because of the orthogonality of the eigenvectors of A. See also , the fastest rotation object References External links Product of Rotations at cut-the-knot. cut-the-knot.org When a Triangle is Equilateral at cut-the-knot. cut-the-knot.org Rotate Points Using Polar Coordinates, howtoproperly.com Rotation in Two Dimensions by Sergio Hannibal Mejia after work by Roger Germundsson and Understanding 3D Rotation by Roger Germundsson, Wolfram Demonstrations Project. demonstrations.wolfram.com Rotation, Reflection, and Frame Change: Orthogonal tensors in computational engineering mechanics, IOP Publishing Euclidean geometry Classical mechanics Orientation (geometry) Kinematics