Question

Check my work please: Complex Eigenvalues/Eigenvectors

So I've taken matrix: \[\left[\begin{matrix}0 & 1 \\ -2 & -2\end{matrix}\right]\]

got the matrix: \[\left[\begin{matrix}1-i & 1 \\ -2 & -1-i\end{matrix}\right]\]
and found my Eigenvalues: \[\lambda = -1\pm i\]
And now to get my Eigenvectors I plug either Eigenvalue into the matrix so, using -1+i, we have: \[\left[\begin{matrix}1-i & 1 \\ -2 & -1-i\end{matrix}\right]\]

Answer 1

This leads to the system of equations \[(1-i)x_1 + x_2 = 0\]
\[-2x_1 + (-1-i)x_2 = 0\]

And where I am stuck. The book gives the solution as \[v_1 = \left(\begin{matrix}1-i \\ 2i\end{matrix}\right)\] and I have no idea how to get that... What am I doing wrong?

Answer 2

note, something went wrong above and step 2 and 3 are out of order.

Answer 3

I've even checked that (1-i)(1-i)+2i = 0 and that -2(1-i)+(-1-i)2i = 0 So clearly the book is right, question is just, what am I missing to come up with those values. Solving it like a normal system of equations doesn't seem to get me anywhere.