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$\forall i,j \in$ $\{$$1, \cdots,n$$\},$ let $x_{i},y_{i}$ be unknowns and $n_{ij} \in \mathbb{Z}$ with $i \le j$ be the knowns. Consider the following $\frac{n(n+1)}{2}$ with $n > 2$ overdetermined bilinear equations: $\sum_{\substack{j=1,}{j \ne i}}^{n} x_{j}y_{j} = n_{ii} -n_{ii} + x_{i}y_{i} \in \mathbb{Z}$. $x_{i}y_{j} + x_{j}y_{i} = n_{ij} \in \mathbb{Z}$ when $i < j$. When is the system solvable and when is it solvable over $\mathbb{Z}$? |
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$\forall i,j \in$ $\{$$1, \cdots,n$$\},$ let $x_{i},y_{i}$ be unknowns and $n_{ij} \in \mathbb{Z}$ with $i \le j$ be the knowns. Consider the following $\frac{n(n+1)}{2}$ with $n > 2$ overdetermined bilinear equations: $\sum_{\substack{j=1,}{j \ne i}}^{n} x_{j}y_{j} = n_{ii} + x_{i}y_{i} \in \mathbb{Z}$. $x_{i}y_{j} + x_{j}y_{i} = n_{ij} \in \mathbb{Z}$ when $i < j$. When is the system solvable and when is it solvable over $\mathbb{Z}$?
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$\forall i,j \in$ $\{$$1, \cdots,n$$\},$ let $x_{i},y_{i}$ be unknowns. Consider the following $\frac{n(n+1)}{2}$ with $n > 2$ overdetermined bilinear equations: $\sum_{\substack{j=1,}{j \ne i}}^{n} x_{i}y_{jx_{j}y_{j} = n_{ii} + x_{i}y_{i} \in \mathbb{Z}$. $x_{i}y_{j} - + x_{j}y_{i} = n_{ij} \in \mathbb{Z}$ when $i < j$. When is the system solvable and when is it solvable over $\mathbb{Z}$? |
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