Posted here too: http://math.stackexchange.com/questions/1711026/two-minimization-problems-using-singular-value-decomposition

Let $q_0, q_1:[0,1]\to \mathbb{R}^n$ be two maps whose components are $L^2[0,1]$, i.e. $q_0, q_1 \in L^2([0,1],\mathbb{R}^n)$. Denote by $||.||$ the Euclidean distance in $\mathbb{R}^n$. Consider the following two minimization problems:

1. minimize $\int_{0}^{1} || q_0(t) - A q_1(t) ||^2 dt $ over $A \in SO(n)$.

2. Consider $U \subset L^2([0,1],\mathbb{R}^n)$ to be the unit sphere in $L^2([0,1],\mathbb{R}^n)$. Let $q_0, q_1 \in U$ now. Then $Aq_1 \in U$ as well for any $A \in SO(n)$. Consider the geodesic $c(A)$ in $U$ joining $q_0, Aq_1$. Minimize the length of $c(A)$.

How can we solve these two problems with/without, preferably with, using singular value decomposition? It was mentioned in the paper http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5601739

But I'd appreciate a detailed explanation for solving these two problems.