xz+yz-2y=0 forma canónica

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

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

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

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

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

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