English: Visualisation 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 σ₁ resp. σ₂ of the semi-major axis resp. semi-minor axis of the ellispe; they are just the singular values which occur as diagonal elements of the scaling Σ.