I derived a following equation: \begin{equation} p_{hl} r^\alpha_{hl} \begin{pmatrix} (p_{i1} r^\alpha_{i1})^{-1} &0&0\\ 0& (p_{j2} r^\alpha_{j2})^{-1}&0\\ 0&0&(p_{k3} r^\alpha_{k3})^{-1}\\ \end{pmatrix}=M^{\alpha}_{hl}, \end{equation} summation over repeating indices is assumed and: \begin{equation} p_{ij} = \begin{pmatrix} p_{11}&p_{12}&p_{13}\\ p_{21}&p_{22}&p_{23}\\ \end{pmatrix}, \:\: r^\alpha_{ij} = \begin{pmatrix} r^\alpha_{11}&r^\alpha_{12}&r^\alpha_{13}\\ r^\alpha_{21}&r^\alpha_{22}&r^\alpha_{23}\\ \end{pmatrix}, \:\: M^\alpha_{ij} = \begin{pmatrix} M^\alpha_{11}&M^\alpha_{12}&M^\alpha_{13}\\ M^\alpha_{21}&M^\alpha_{22}&M^\alpha_{23}\\ \end{pmatrix}, \end{equation} where $\alpha = \{1,2,3,4\}$. The equation look not very pretty but its geometric meaning is simple: I have 3 2d vectors defined by $p_{ij}$. In order to match a given set of 3 normalized vectors defined by $M^{\alpha}_{ij}$. Different $M^{\alpha}_{ij}$ tune each coordinate by a factor $r^{\alpha}_{ij}$ (it is is mostly matrix of ones). Matrix after $p_{hl} r^\alpha_{hl}$ normalizes vectors. In tensor notations, this equation looks a bit simpler: \begin{equation} \sum_i (\delta_{ih}-M^{\alpha}_{hl})p_{il}r^{\alpha}_{il} = 0 \end{equation}

There exists a unique solution $p_{ij}$. My interest is in $r^{\alpha}_{ij}$ for a given $r^{\beta}_{ij}$. I subtract (add) the above equation and the one with $\alpha \to \beta$. As a result, I have two equations for $\alpha\neq\beta\in\{1,2,3,4\}$: \begin{equation} \sum_i \left(M^\beta_{hl} r^\beta_{il}-M^\alpha_{hl} r^\alpha_{il}\right)p_{il}=0, \:\:\:\: \sum_i \left((\delta_{ih}-M^\alpha_{hl}) r^\alpha_{il} +(\delta_{ih}-M^\beta_{hl}) r^\beta_{il}\right)p_{il}=0. \end{equation}

I can map $2\times3$ tensors on $6d$ vectors and deal with two homogeneous linear systems of equations. Non-trivial solutions condition results in 2 algebraic equations of 6th power. I am interested in minimum number of $\{r^{\alpha}_{ij},r^\beta_{ij}\}\neq1$, so I expect to factorize $(r^{\alpha}_{ij}-1)$ and $(r^{\beta}_{ij}-1)$ from those polynomials.

Am I missing a better parametrization to avoid some of the hardships of solving high-order algebraic equations?



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