As we know, the general linear mapping send the unit n-sphere into a n-ellipsoid.
SVD decomposes this action into a sequence 1.rotation/reflection -- 2.shearing -- 3.rotation/reflection.
The first question is: why two rotoreflections? As far I imagine, any action sending the unit spere into an ellipsoid should be decomposed as product of two transformations:
Simply shear the sphere, then rotate/reflect.
Second question has to do with the operator A*A. Every symmetric operator carries a bilinear n-form (that describes an n-surface). What this form has to do with SVD decomposition of A?
Thanks.
In the SVD $$ A = U D V^*,$$
the "stretching" matrix $D$ is diagonal - this means that not only is it a stretching of the sphere into an ellipsoid, but it is one where the principal axes of the stretching are the cartesian coordinate axes. For a general stretching of the sphere into an ellipsoid these axes may not be aligned, which necessitates the rotation/reflection $V^*$.
In terms of linear transformations, it is entirely correct to look at it the way you suggest - by choosing the orthonormal basis made up by the columns of $V$, the transformation now has the matrix $$ V^* A V = (V^* U) D,$$ which is a stretching followed by a rotation/reflection as you describe.
Regarding the bilinear form corresponding to $A^*A$: it acts as$$(A^*A)(u,v) = u^* A^* A v = (Au)^*(Av) = \langle Au, Av \rangle;$$ so it is the pullback of the inner product on the codomain by the map $A$. Thus if $u$ is a singular vector (principal axis) with singular value (stretching factor) $\lambda$ we have $$(A^*A)(u,v) = \lambda^2 \langle u,v \rangle$$ for every $v$.