文件:Singular value decomposition.gif

Animated illustration of the SVD of a 2D, real shearing matrix [math]\displaystyle{ M }[/math]. First, we see the unit disc in blue together with the two canonical unit vectors. We then see the actions of [math]\displaystyle{ M }[/math], which distorts the disk to an ellipse. The SVD decomposes [math]\displaystyle{ M }[/math] into three simple transformations: an initial rotation [math]\displaystyle{ V^⁎ }[/math], a scaling [math]\displaystyle{ {\displaystyle \mathbf {\Sigma } } }[/math] along the coordinate axes, and a final rotation U. The lengths [math]\displaystyle{ σ_1 }[/math] and [math]\displaystyle{ σ_2 }[/math] of the semi-axes of the ellipse are the singular values of [math]\displaystyle{ M }[/math], namely [math]\displaystyle{ Σ_(1,1) }[/math] and [math]\displaystyle{ _Σ(2,2) }[/math].