Editing Eigenvalue algorithm (section)

==== Eigenvectors of normal 3×3 matrices ====
If a 3×3 matrix <math>A</math> is normal, then the cross-product can be used to find eigenvectors. If <math>\lambda</math> is an eigenvalue of <math>A</math>, then the null space of <math>A - \lambda I</math> is perpendicular to its column space. The [[cross product]] of two independent columns of  <math>A - \lambda I</math> will be in the null space. That is, it will be an eigenvector associated with <math>\lambda</math>. Since the column space is two dimensional in this case, the eigenspace must be one dimensional, so any other eigenvector will be parallel to it.

If <math>A - \lambda I</math> does not contain two independent columns but is not {{math|'''0'''}}, the cross-product can still be used. In this case <math>\lambda</math> is an eigenvalue of multiplicity 2, so any vector perpendicular to the column space will be an eigenvector. Suppose <math>\mathbf v</math> is a non-zero column of <math>A - \lambda I</math>. Choose an arbitrary vector <math>\mathbf u</math> not parallel to <math>\mathbf v</math>. Then <math>\mathbf v\times \mathbf u</math>  and  <math>(\mathbf v\times \mathbf u)\times \mathbf v</math> will be perpendicular to <math>\mathbf v</math> and thus will be eigenvectors of  <math>\lambda</math>.

This does not work when <math>A</math> is not normal, as the null space and column space do not need to be perpendicular for such matrices.