I am learning about complex eigenvalues in Linear Algebra and I am confused with one problem. I have a matrix in $A-\lambda I $ form. For the eigenvalue $\lambda=3+2i$, $A-\lambda I=\begin{bmatrix} 2-2i & 1\\ -8 & -2-2i \end{bmatrix}$ and for the eigenvalue $\lambda=3-2i$, $A-\lambda I=\begin{bmatrix} 2+2i & 1\\ -8 & -2+2i \end{bmatrix}$. I am supposed to find a basis for each eigenspace in $\mathbb{C}^2$ . I know that to find a basis for the eigenspace, I need to find the solutions to the homogeneous system. For $\lambda=3+2i$, the system of equations is $(2-2i)x_{1}+1x_{2}=0$ and $-8x_{1}+(-2-2i)x_{2}=0$.
This is where I am getting confused. In the example, the book states that both of the equations above determine the same relationship between $x_{1}$ and $x_{2}$, and either equation can be used to express one variable in terms of the other.
Using that fact, I should be able to find the basis for the eigenspace by using one of the equations because both equations should give the same relationship between the two variables. I have solved the second equation $-8x_{1}+(-2-2i)x_{2}=0$ and found that the relationship was $x_{1}=(-\frac{1}{4}-\frac{1}{4}i)x_{2}$. This equation gave me the same basis as the answer in the back of my book.
However when I try to use the first equation $(2-2i)x_{1}+1x_{2}=0$, I do not get the same relationship as the second equation. I got $x_{1}=(-\frac{1}{2}+\frac{1}{2}i)x_{2}$. I do not know why the second equation is off by a factor of $1/2$, according to the book both equations should give me the same relationship between the two variables. How come the second equation does not give me the same relationship as the first equation between the two variables?
Look at your first equation:
$ (2-2i)x_1 + x_2 = 0 $
$ x_1 = -\frac{x_2}{2-2i}$
$ x_1 = -\frac{x_2(2+2i)}{(2-2i)(2+2i)}$
$x_1 = \Big(-\frac{1}{4} - \frac{1}{4}i\Big)x_2$