On page 155 of Tristan Needham's Visual Differential Geometry and Forms, the singular value decomposition (SVD) is given by
$$M = R_{\phi} \circ \Sigma \circ R_{-\theta}$$
with the associated picture:
I am a bit confused as to why we actually rotate by $-\theta$ , expand then rotate back? Couldn't we just expand then rotate so the circle to the ellipse?