SVD Example

Consider the linear transformation $T: \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ given by the matrix

A = \left[ \begin{array}{cc} 2 & 1 \\ 0 & 1 \end{array} \right].

The unit circle is mapped to the blue ellipse you see below.

You can see the geometric enterpretation of the columns of $A$ as the images of the standard basis vectors. The diagram below shows $e_1$ and $T(e_1)$ first. \\ Press the $e_2$ button to see $e_2$ and $T(e_2)$ .

Drag the point around the unit circle to see where T sends it.

Figure 1: e1 and T(e1)

You can see that although $(e_1, e_2)$ is an orthonormal basis for the domain, $(T(e_1), T(e_2))$ is not. They are not even perpendicular.

Click on $v_1$ and $v_2$ on the figure. They are both on the unit circle and are at right angles from each other. You will see that their images yield the major and minor axes of the ellipse that comprises the image of the unit circle. You can also see that both axes are rotated about $18.7^\circ$ for their transformation.