Transformation matrix to go from one vector to another

Question

I've two vectors $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$. How to find transformation matrix for transform from a to b?

Answer 1

Try the dyadic product $$ \mathbf{a b} \equiv \mathbf{a}\otimes\mathbf{b} \equiv \mathbf{a b}^\mathrm{T} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\begin{pmatrix} b_1 & b_2 & b_3 \end{pmatrix} = \begin{pmatrix} a_1b_1 & a_1b_2 & a_1b_3 \\ a_2b_1 & a_2b_2 & a_2b_3 \\ a_3b_1 & a_3b_2 & a_3b_3 \end{pmatrix}. $$ Maybe you should include a normalization factor.

Answer 2

If the $a_i$ are non-zero you can take the diagonal matrix with entries $a_{ii}=\frac{b_i}{a_i}$. If $a_1=0$ then replace $a_{11}$ with zero and take $a_{12}=\frac{b_1}{a_1}$. Similarly if $a_2$ or $a_3$ is zero.