English: Visual representation of a singular value decomposition of a 2-dimensional, real
shearing M. First, we see the unit disc in blue together with the two canonical unit vectors. We then see the action of
M on the unit disc: it distorts the circle to an ellipse. The SVD decomposes
M into three simple transformations: a rotation
V*, a scaling Σ along the coordinate axes and a second rotation
U. The SVD reveals the rotation of the ellipse with respect to the coordinate axes and the lengths σ
1 resp. σ
2 of the
semi-major axis resp.
semi-minor axis of the ellipse; they are just the singular values which occur as diagonal elements of the scaling Σ.