I would like to reduce such equations. Is there a general method that could help me to understnad how to do so. $$Q_1= xy + yz + zx + 2y + 1 = 0 \\ Q_2 = x^2+y^2+z^2-4xy -4xz-4yz-4z-4y+2x+1=0$$
This is what I tried : $$\begin{align*} Q_1& =xy + yz + zx + 2y + 1 \\&=(x+z)(y+z)-z^2 +2y + 1 \\ &=\frac{1}{4}(x+y+2z)^2 -\frac{1}{4}(x-y)^2 -z^2 + 2y +1 \\ \end{align*}$$ but after that I don't how to continue the reduction.
For the second equation : $$ \begin{align*} Q_2 &= x^2+y^2+z^2-4xy -4xz-4yz-4z-4y+2x+1 \\ &= (x-2y-2z)^2 - 3y^2-3z^2-4z-4y+2x+1\end{align*}$$ I don't think it is right.
matrix work for your first polynomial, after making homogeneous with a new variable $t,$ where $H$ is the Hessian matrix of the resulting form. The linear expressions that are squared in the "reduction" have coefficients in the rows of $Q.$ The entries of $D$ give the coefficients for double the original form, so you need to be careful, divide the $D$ numbers by two, and check the results.
$$ Q^T D Q = H $$ $$\left( \begin{array}{rrrr} 0 & - \frac{ 1 }{ 2 } & 3 & 1 \\ 1 & 1 & 0 & 0 \\ 0 & - \frac{ 1 }{ 2 } & 1 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right) \left( \begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & - 2 & 0 & 0 \\ 0 & 0 & \frac{ 1 }{ 2 } & 0 \\ 0 & 0 & 0 & - 4 \\ \end{array} \right) \left( \begin{array}{rrrr} 0 & 1 & 0 & 1 \\ - \frac{ 1 }{ 2 } & 1 & - \frac{ 1 }{ 2 } & 0 \\ 3 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right) = \left( \begin{array}{rrrr} 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 2 \\ 1 & 1 & 0 & 0 \\ 0 & 2 & 0 & 2 \\ \end{array} \right) $$
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Algorithm discussed at http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr
https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia
$$ H = \left( \begin{array}{rrrr} 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 2 \\ 1 & 1 & 0 & 0 \\ 0 & 2 & 0 & 2 \\ \end{array} \right) $$ $$ D_0 = H $$ $$ E_j^T D_{j-1} E_j = D_j $$ $$ P_{j-1} E_j = P_j $$ $$ E_j^{-1} Q_{j-1} = Q_j $$ $$ P_j Q_j = Q_j P_j = I $$ $$ P_j^T H P_j = D_j $$ $$ Q_j^T D_j Q_j = H $$
$$ H = \left( \begin{array}{rrrr} 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 2 \\ 1 & 1 & 0 & 0 \\ 0 & 2 & 0 & 2 \\ \end{array} \right) $$
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$$ E_{1} = \left( \begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right) $$ $$ P_{1} = \left( \begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right) , \; \; \; Q_{1} = \left( \begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right) , \; \; \; D_{1} = \left( \begin{array}{rrrr} 2 & 2 & 0 & 0 \\ 2 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ \end{array} \right) $$
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$$ E_{2} = \left( \begin{array}{rrrr} 1 & - 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{2} = \left( \begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & - 1 & 0 & 0 \\ \end{array} \right) , \; \; \; Q_{2} = \left( \begin{array}{rrrr} 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right) , \; \; \; D_{2} = \left( \begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & - 2 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ \end{array} \right) $$
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$$ E_{3} = \left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & \frac{ 1 }{ 2 } & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{3} = \left( \begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & \frac{ 1 }{ 2 } & 0 \\ 0 & 0 & 1 & 0 \\ 1 & - 1 & - \frac{ 1 }{ 2 } & 0 \\ \end{array} \right) , \; \; \; Q_{3} = \left( \begin{array}{rrrr} 0 & 1 & 0 & 1 \\ 0 & 1 & - \frac{ 1 }{ 2 } & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right) , \; \; \; D_{3} = \left( \begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & - 2 & 0 & 1 \\ 0 & 0 & \frac{ 1 }{ 2 } & \frac{ 3 }{ 2 } \\ 0 & 1 & \frac{ 3 }{ 2 } & 0 \\ \end{array} \right) $$
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$$ E_{4} = \left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & \frac{ 1 }{ 2 } \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{4} = \left( \begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & \frac{ 1 }{ 2 } & \frac{ 1 }{ 2 } \\ 0 & 0 & 1 & 0 \\ 1 & - 1 & - \frac{ 1 }{ 2 } & - \frac{ 1 }{ 2 } \\ \end{array} \right) , \; \; \; Q_{4} = \left( \begin{array}{rrrr} 0 & 1 & 0 & 1 \\ - \frac{ 1 }{ 2 } & 1 & - \frac{ 1 }{ 2 } & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right) , \; \; \; D_{4} = \left( \begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & - 2 & 0 & 0 \\ 0 & 0 & \frac{ 1 }{ 2 } & \frac{ 3 }{ 2 } \\ 0 & 0 & \frac{ 3 }{ 2 } & \frac{ 1 }{ 2 } \\ \end{array} \right) $$
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$$ E_{5} = \left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & - 3 \\ 0 & 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{5} = \left( \begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & \frac{ 1 }{ 2 } & - 1 \\ 0 & 0 & 1 & - 3 \\ 1 & - 1 & - \frac{ 1 }{ 2 } & 1 \\ \end{array} \right) , \; \; \; Q_{5} = \left( \begin{array}{rrrr} 0 & 1 & 0 & 1 \\ - \frac{ 1 }{ 2 } & 1 & - \frac{ 1 }{ 2 } & 0 \\ 3 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right) , \; \; \; D_{5} = \left( \begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & - 2 & 0 & 0 \\ 0 & 0 & \frac{ 1 }{ 2 } & 0 \\ 0 & 0 & 0 & - 4 \\ \end{array} \right) $$
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$$ P^T H P = D $$ $$\left( \begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & - 1 \\ 0 & \frac{ 1 }{ 2 } & 1 & - \frac{ 1 }{ 2 } \\ 1 & - 1 & - 3 & 1 \\ \end{array} \right) \left( \begin{array}{rrrr} 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 2 \\ 1 & 1 & 0 & 0 \\ 0 & 2 & 0 & 2 \\ \end{array} \right) \left( \begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & \frac{ 1 }{ 2 } & - 1 \\ 0 & 0 & 1 & - 3 \\ 1 & - 1 & - \frac{ 1 }{ 2 } & 1 \\ \end{array} \right) = \left( \begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & - 2 & 0 & 0 \\ 0 & 0 & \frac{ 1 }{ 2 } & 0 \\ 0 & 0 & 0 & - 4 \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrrr} 0 & - \frac{ 1 }{ 2 } & 3 & 1 \\ 1 & 1 & 0 & 0 \\ 0 & - \frac{ 1 }{ 2 } & 1 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right) \left( \begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & - 2 & 0 & 0 \\ 0 & 0 & \frac{ 1 }{ 2 } & 0 \\ 0 & 0 & 0 & - 4 \\ \end{array} \right) \left( \begin{array}{rrrr} 0 & 1 & 0 & 1 \\ - \frac{ 1 }{ 2 } & 1 & - \frac{ 1 }{ 2 } & 0 \\ 3 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right) = \left( \begin{array}{rrrr} 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 2 \\ 1 & 1 & 0 & 0 \\ 0 & 2 & 0 & 2 \\ \end{array} \right) $$