2*x^2+2*y^2+2*z^2+2*x*y-2*x*z-2*y*z-14*x-14*y+4*z=26 forma canónica

Se da la ecuación de superficie de 2 grado:
$$2 x^{2} + 2 x y - 2 x z - 14 x + 2 y^{2} - 2 y z - 14 y + 2 z^{2} + 4 z - 26 = 0$$
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
$$a_{11} = 2$$
$$a_{12} = 1$$
$$a_{13} = -1$$
$$a_{14} = -7$$
$$a_{22} = 2$$
$$a_{23} = -1$$
$$a_{24} = -7$$
$$a_{33} = 2$$
$$a_{34} = 2$$
$$a_{44} = -26$$
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
$$I_{1} = 6$$

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

$$I_{3} = \left|\begin{matrix}2 & 1 & -1\\1 & 2 & -1\\-1 & -1 & 2\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}2 & 1 & -1 & -7\\1 & 2 & -1 & -7\\-1 & -1 & 2 & 2\\-7 & -7 & 2 & -26\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}2 - \lambda & 1 & -1\\1 & 2 - \lambda & -1\\-1 & -1 & 2 - \lambda\end{matrix}\right|$$

     |2   -7 |   |2   -7 |   |2   2 |
K2 = |       | + |       | + |      |
     |-7  -26|   |-7  -26|   |2  -26|

     |2   1   -7 |   |2   -1  -7 |   |2   -1  -7 |
     |           |   |           |   |           |
K3 = |1   2   -7 | + |-1  2    2 | + |-1  2    2 |
     |           |   |           |   |           |
     |-7  -7  -26|   |-7  2   -26|   |-7  2   -26|

$$I_{1} = 6$$
$$I_{2} = 9$$
$$I_{3} = 4$$
$$I_{4} = -256$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 6 \lambda^{2} - 9 \lambda + 4$$
$$K_{2} = -258$$
$$K_{3} = -488$$
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
$$\lambda^{3} - 6 \lambda^{2} + 9 \lambda - 4 = 0$$
$$\lambda_{1} = 4$$
$$\lambda_{2} = 1$$
$$\lambda_{3} = 1$$
entonces la forma canónica de la ecuación será
$$\left(\tilde z^{2} \lambda_{3} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right)\right) + \frac{I_{4}}{I_{3}} = 0$$
$$4 \tilde x^{2} + \tilde y^{2} + \tilde z^{2} - 64 = 0$$

        2           2           2    
\tilde x    \tilde y    \tilde z     
--------- + --------- + --------- = 1
        2         2           2      
 /  1  \      / 1\        / 1\       
 |-----|      \8 /        \8 /       
 \2*1/8/                             

es la ecuación para el tipo elipsoide
- está reducida a la forma canónica