Editing Singular value decomposition (section)

=== Singular values as semiaxes of an ellipse or ellipsoid ===
As shown in the figure, the [[singular values]] can be interpreted as the magnitude of the semiaxes of an [[ellipse]] in 2D. This concept can be generalized to {{tmath|n}}-dimensional [[Euclidean space]], with the singular values of any {{tmath|n \times n}} [[square matrix]] being viewed as the magnitude of the semiaxis of an {{tmath|n}}-dimensional [[ellipsoid]]. Similarly, the singular values of any {{tmath|m \times n}} matrix can be viewed as the magnitude of the semiaxis of an {{tmath|n}}-dimensional [[ellipsoid]] in {{tmath|m}}-dimensional space, for example as an ellipse in a (tilted) 2D plane in a 3D space. Singular values encode magnitude of the semiaxis, while singular vectors encode direction.  See [[#Geometric meaning|below]] for further details.