Given line equation of 2-order:
$$x_{1}^{2} + 4 x_{2}^{2} = 0$$
This equation looks like:
$$a_{11} x_{2}^{2} + 2 a_{12} x_{1} x_{2} + a_{22} x_{1}^{2} + 2 a_{13} x_{2} + 2 a_{23} x_{1} + a_{33} = 0$$
where
$$a_{11} = 4$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{22} = 1$$
$$a_{23} = 0$$
$$a_{33} = 0$$
The invariants of the equation when converting coordinates are determinants:
$$I_{1} = a_{11} + a_{22}$$
|a11 a12|
I2 = | |
|a12 a22|
$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12}\\a_{12} & a_{22} - \lambda\end{matrix}\right|$$
|a11 a13| |a22 a23|
K2 = | | + | |
|a13 a33| |a23 a33|
substitute coefficients
$$I_{1} = 5$$
|4 0|
I2 = | |
|0 1|
$$I_{3} = \left|\begin{matrix}4 & 0 & 0\\0 & 1 & 0\\0 & 0 & 0\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}- \lambda + 4 & 0\\0 & - \lambda + 1\end{matrix}\right|$$
|4 0| |1 0|
K2 = | | + | |
|0 0| |0 0|
$$I_{1} = 5$$
$$I_{2} = 4$$
$$I_{3} = 0$$
$$I{\left(\lambda \right)} = \lambda^{2} - 5 \lambda + 4$$
$$K_{2} = 0$$
Because
$$I_{3} = 0 \wedge I_{2} > 0$$
then by line type:
this equation is of type : degenerate ellipse
Make the characteristic equation for the line:
$$- I_{1} \lambda + \lambda^{2} + I_{2} = 0$$
or
$$\lambda^{2} - 5 \lambda + 4 = 0$$
Solve this equation$$\lambda_{1} = 4$$
$$\lambda_{2} = 1$$
then the canonical form of the equation will be
$$\tilde x1^{2} \lambda_{2} + \tilde x2^{2} \lambda_{1} + \frac{I_{3}}{I_{2}} = 0$$
or
$$\tilde x1^{2} + 4 \tilde x2^{2} = 0$$
$$\frac{\tilde x1^{2}}{1^{2}} + \frac{\tilde x2^{2}}{\left(\frac{1}{2}\right)^{2}} = 0$$
- reduced to canonical form