4z-yz-3z=12 forma canónica

Se da la ecuación de superficie de 2 grado:
$$- y z + z - 12 = 0$$
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
$$a_{11} = 0$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{14} = 0$$
$$a_{22} = 0$$
$$a_{23} = - \frac{1}{2}$$
$$a_{24} = 0$$
$$a_{33} = 0$$
$$a_{34} = \frac{1}{2}$$
$$a_{44} = -12$$
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
$$I_{1} = 0$$

     |0  0|   | 0    -1/2|   |0  0|
I2 = |    | + |          | + |    |
     |0  0|   |-1/2   0  |   |0  0|

$$I_{3} = \left|\begin{matrix}0 & 0 & 0\\0 & 0 & - \frac{1}{2}\\0 & - \frac{1}{2} & 0\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & - \frac{1}{2} & 0\\0 & - \frac{1}{2} & 0 & \frac{1}{2}\\0 & 0 & \frac{1}{2} & -12\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}- \lambda & 0 & 0\\0 & - \lambda & - \frac{1}{2}\\0 & - \frac{1}{2} & - \lambda\end{matrix}\right|$$

     |0   0 |   |0   0 |   | 0   1/2|
K2 = |      | + |      | + |        |
     |0  -12|   |0  -12|   |1/2  -12|

     |0  0   0 |   | 0    -1/2   0 |   |0   0    0 |
     |         |   |               |   |           |
K3 = |0  0   0 | + |-1/2   0    1/2| + |0   0   1/2|
     |         |   |               |   |           |
     |0  0  -12|   | 0    1/2   -12|   |0  1/2  -12|

$$I_{1} = 0$$
$$I_{2} = - \frac{1}{4}$$
$$I_{3} = 0$$
$$I_{4} = 0$$
$$I{\left(\lambda \right)} = - \lambda^{3} + \frac{\lambda}{4}$$
$$K_{2} = - \frac{1}{4}$$
$$K_{3} = 3$$
Como
$$I_{3} = 0 \wedge I_{4} = 0 \wedge I_{2} \neq 0$$
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
$$\lambda^{3} - \frac{\lambda}{4} = 0$$
$$\lambda_{1} = - \frac{1}{2}$$
$$\lambda_{2} = \frac{1}{2}$$
$$\lambda_{3} = 0$$
entonces la forma canónica de la ecuación será
$$\left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) + \frac{K_{3}}{I_{2}} = 0$$
$$- \frac{\tilde x^{2}}{2} + \frac{\tilde y^{2}}{2} - 12 = 0$$
$$\frac{\tilde x^{2}}{24} - \frac{\tilde y^{2}}{24} = -1$$
es la ecuación para el tipo cilindro hiperbólico
- está reducida a la forma canónica