Finding the matrix which generate a vector series

Question

We have vectors $v_0,\ldots,v_k$ in $\Bbb R^n$. Find the matrix A which minimizes the $\sum_{i=0}^{k-1}|Av_i-v_{i+1}|$.

Answer 1

You can write the minimizing problem $$\min_A\sum_{i=0}^{k-1}\|Av_i-v_{i+1}\|,$$ as $$\begin{cases} \min_A\quad \sum_{i=0}^{k-1} t_i \\ s.t\,\,\,\,\,\, \quad \|Av_i-v_{i+1}\|^2 = t_i^2 \end{cases} .$$

If you see the matrix $A$ as a long vector, you can try to solve it umerically with constrOptim tool, for instance.

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