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How do I convert the 3-variable quadratic form q(x₁ x₂ x₃) = row vector [x₁ x₂ x₃] multiplied by [a₁₁ a₁₂ a₁₃ a₂₁ a₂₂ a₂₃ a₃₁ a₃₂ a₃₃]₃ₓ₃ matrix multiplied by column vector [x₁ x₂ x₃] into the equation: $$ |A_1|\left(x_1 + x_2 a_{12}/a_{11} + x_3 a_{13}/a_{11}\right)^2 + (|A₂|/|A₁|)\left(x₂ + x₃(a₁₁a₂₃ - a₁₂a₁₃)/(a₁₁a₁₂ - a²₁₂)\right))^2 + (|A₃|/|A₂|)x^2_3 $$ by "completing the square" method? Please be as detailed as possible, it's doing my head in... thank you.

$$ \left[ \begin{array}{ccc} x_1 & x_2 & x_3 \end{array} \right] \left[ \begin{array}{ccc} a_{11}& a_{12} & a_{13} \\ a_{21}& a_{22} & a_{23} \\ a_{31}& a_{32} & a_{33} \\ \end{array} \right] \left[ \begin{array}{ccc} x_1 \\ x_2 \\ x_3 \end{array} \right] $$

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  • $\begingroup$ Please typeset the question in an intelligible way. $\endgroup$ – Yves Daoust Jun 2 at 8:20
  • $\begingroup$ By the way, what are $|A_1|, |A_2| $ ? $\endgroup$ – Matti P. Jun 2 at 8:22
  • $\begingroup$ @MattiP. Hey thank you so much for editing my question, I'm really new to this thing. |A₁| is a₁₁ and |A₂| is the determinant of [a₁₁ a₂₁ a ₁₂ a₂₂] $\endgroup$ – Francis Scott Jun 2 at 13:19
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    $\begingroup$ @YvesDaoust Thank you so much for the hint, I really appreciate it. I've just joined the website and I'm gonna get the hang of inputting everything properly with time $\endgroup$ – Francis Scott Jun 2 at 13:21
  • $\begingroup$ I discuss an algorithm for symmetric matrices at math.stackexchange.com/questions/1388421/… Eventually I saw Lagrange's method in Gantmacher's book, Lagrange had a very specific step when all remaining diagonal entries were zero... $\endgroup$ – Will Jagy Jun 2 at 19:15
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Hint:

$$ax^2+2bxy+2cxz+\cdots=a\left(x+\frac bay+\frac caz\right)^2-\frac{b^2}ay^2-\frac{c^2}az^2+\cdots$$

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