Given equation of the surface of 2-order:
$$x_{1}^{2} + 4 x_{1} x_{2} + 2 x_{1} x_{3} + x_{2}^{2} + 2 x_{2} x_{3} + 3 x_{3}^{2} = 0$$
This equation looks like:
$$a_{11} x_{3}^{2} + 2 a_{12} x_{2} x_{3} + 2 a_{13} x_{1} x_{3} + 2 a_{14} x_{3} + a_{22} x_{2}^{2} + 2 a_{23} x_{1} x_{2} + 2 a_{24} x_{2} + a_{33} x_{1}^{2} + 2 a_{34} x_{1} + a_{44} = 0$$
where
$$a_{11} = 3$$
$$a_{12} = 1$$
$$a_{13} = 1$$
$$a_{14} = 0$$
$$a_{22} = 1$$
$$a_{23} = 2$$
$$a_{24} = 0$$
$$a_{33} = 1$$
$$a_{34} = 0$$
$$a_{44} = 0$$
The invariants of the equation when converting coordinates are determinants:
$$I_{1} = a_{11} + a_{22} + a_{33}$$
|a11 a12| |a22 a23| |a11 a13|
I2 = | | + | | + | |
|a12 a22| |a23 a33| |a13 a33|
$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\a_{12} & a_{22} & a_{23} & a_{24}\\a_{13} & a_{23} & a_{33} & a_{34}\\a_{14} & a_{24} & a_{34} & a_{44}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12} & a_{13}\\a_{12} & a_{22} - \lambda & a_{23}\\a_{13} & a_{23} & a_{33} - \lambda\end{matrix}\right|$$
|a11 a14| |a22 a24| |a33 a34|
K2 = | | + | | + | |
|a14 a44| |a24 a44| |a34 a44|
|a11 a12 a14| |a22 a23 a24| |a11 a13 a14|
| | | | | |
K3 = |a12 a22 a24| + |a23 a33 a34| + |a13 a33 a34|
| | | | | |
|a14 a24 a44| |a24 a34 a44| |a14 a34 a44|
substitute coefficients
$$I_{1} = 5$$
|3 1| |1 2| |3 1|
I2 = | | + | | + | |
|1 1| |2 1| |1 1|
$$I_{3} = \left|\begin{matrix}3 & 1 & 1\\1 & 1 & 2\\1 & 2 & 1\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}3 & 1 & 1 & 0\\1 & 1 & 2 & 0\\1 & 2 & 1 & 0\\0 & 0 & 0 & 0\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}3 - \lambda & 1 & 1\\1 & 1 - \lambda & 2\\1 & 2 & 1 - \lambda\end{matrix}\right|$$
|3 0| |1 0| |1 0|
K2 = | | + | | + | |
|0 0| |0 0| |0 0|
|3 1 0| |1 2 0| |3 1 0|
| | | | | |
K3 = |1 1 0| + |2 1 0| + |1 1 0|
| | | | | |
|0 0 0| |0 0 0| |0 0 0|
$$I_{1} = 5$$
$$I_{2} = 1$$
$$I_{3} = -7$$
$$I_{4} = 0$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 5 \lambda^{2} - \lambda - 7$$
$$K_{2} = 0$$
$$K_{3} = 0$$
Because
I3 != 0
then by type of surface:
you need to
Make the characteristic equation for the surface:
$$- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0$$
or
$$\lambda^{3} - 5 \lambda^{2} + \lambda + 7 = 0$$
$$\lambda_{1} = -1$$
$$\lambda_{2} = 3 - \sqrt{2}$$
$$\lambda_{3} = \sqrt{2} + 3$$
then the canonical form of the equation will be
$$\left(\tilde x1^{2} \lambda_{3} + \left(\tilde x2^{2} \lambda_{2} + \tilde x3^{2} \lambda_{1}\right)\right) + \frac{I_{4}}{I_{3}} = 0$$
$$\tilde x1^{2} \left(\sqrt{2} + 3\right) + \tilde x2^{2} \left(3 - \sqrt{2}\right) - \tilde x3^{2} = 0$$
$$- \frac{\tilde x3^{2}}{1^{2}} + \left(\frac{\tilde x1^{2}}{\left(\frac{1}{\sqrt{\sqrt{2} + 3}}\right)^{2}} + \frac{\tilde x2^{2}}{\left(\frac{1}{\sqrt{3 - \sqrt{2}}}\right)^{2}}\right) = 0$$
this equation is fora type cone
- reduced to canonical form