x^2-4*x*y+4*x*z-2*y^2+8*y*z-2*z^2 forma canónica

Se da la ecuación de superficie de 2 grado:
$$x^{2} - 4 x y + 4 x z - 2 y^{2} + 8 y z - 2 z^{2} = 0$$
Esta ecuación tiene la forma:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} y z + 2 a_{24} y + a_{33} z^{2} + 2 a_{34} z + a_{44} = 0$$
donde
$$a_{11} = 1$$
$$a_{12} = -2$$
$$a_{13} = 2$$
$$a_{14} = 0$$
$$a_{22} = -2$$
$$a_{23} = 4$$
$$a_{24} = 0$$
$$a_{33} = -2$$
$$a_{34} = 0$$
$$a_{44} = 0$$
Las invariantes de esta ecuación al transformar las coordenadas son los determinantes:
$$I_{1} = a_{11} + a_{22} + a_{33}$$

     |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|

sustituimos coeficientes
$$I_{1} = -3$$

     |1   -2|   |-2  4 |   |1  2 |
I2 = |      | + |      | + |     |
     |-2  -2|   |4   -2|   |2  -2|

$$I_{3} = \left|\begin{matrix}1 & -2 & 2\\-2 & -2 & 4\\2 & 4 & -2\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}1 & -2 & 2 & 0\\-2 & -2 & 4 & 0\\2 & 4 & -2 & 0\\0 & 0 & 0 & 0\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}1 - \lambda & -2 & 2\\-2 & - \lambda - 2 & 4\\2 & 4 & - \lambda - 2\end{matrix}\right|$$

     |1  0|   |-2  0|   |-2  0|
K2 = |    | + |     | + |     |
     |0  0|   |0   0|   |0   0|

     |1   -2  0|   |-2  4   0|   |1  2   0|
     |         |   |         |   |        |
K3 = |-2  -2  0| + |4   -2  0| + |2  -2  0|
     |         |   |         |   |        |
     |0   0   0|   |0   0   0|   |0  0   0|

$$I_{1} = -3$$
$$I_{2} = -24$$
$$I_{3} = -28$$
$$I_{4} = 0$$
$$I{\left(\lambda \right)} = - \lambda^{3} - 3 \lambda^{2} + 24 \lambda - 28$$
$$K_{2} = 0$$
$$K_{3} = 0$$
Como

I3 != 0

entonces por razón de tipos de rectas:
hay que
Formulamos la ecuación característica para nuestra superficie:
$$- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0$$
o
$$\lambda^{3} + 3 \lambda^{2} - 24 \lambda + 28 = 0$$
$$\lambda_{1} = -7$$
$$\lambda_{2} = 2$$
$$\lambda_{3} = 2$$
entonces la forma canónica de la ecuación será
$$\left(\tilde z^{2} \lambda_{3} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right)\right) + \frac{I_{4}}{I_{3}} = 0$$
$$- 7 \tilde x^{2} + 2 \tilde y^{2} + 2 \tilde z^{2} = 0$$
$$- \frac{\tilde x^{2}}{\left(\frac{\sqrt{7}}{7}\right)^{2}} + \left(\frac{\tilde y^{2}}{\left(\frac{\sqrt{2}}{2}\right)^{2}} + \frac{\tilde z^{2}}{\left(\frac{\sqrt{2}}{2}\right)^{2}}\right) = 0$$
es la ecuación para el tipo cono
- está reducida a la forma canónica