eigenvalues and eigenvectors: I'm trying out examples out of my course for DE system solution methods. But it seems to me the course got it wrong in an example when it comes to the calculation of eigenvectors. Detailed information follows.

Question

anonymous · Answer

$$\left[\begin{matrix}1 &-1\ 5 & -3\end{matrix}\right]$$
the eigenvalue equation is $$r ^{2}+2r+2=0$$
$$\rightarrow r \in \left\{ -1+i, -1-i \right\}$$

This far I agree and get the same eigenvalues.
Now when I try to get the eigenvector for -1+i, I end up with the following system

$$\left[\begin{matrix}1 & -1 \ 5 & -3\end{matrix}\right]-\left[\begin{matrix}-1+i &0 \ 0 & -1+i\end{matrix}\right]=\left[\begin{matrix}2-i & -1\ 5 & 4-i\end{matrix}\right]=\left[\begin{matrix}2-i & -1 \ 0 & 6\end{matrix}\right]$$

The course claims the eigenvector is (1,2-i).

I can only get that result if I drop the second row of the system. But the second row of the matrix would indicate that 6 .v2 = -1+i and so v2 must be (-1+i)/6. It seems to me there is a problem to determine the eigenvectors, because of conflict.

zzr0ck3r · Answer

i am confused by this

$\left[\begin{matrix}2-i & -1\ 5 & 4-i\end{matrix}\right]=\left[\begin{matrix}2-i & -1 \ 0 & 6\end{matrix}\right]$

anonymous · Answer

but the entry 22 is not 4-i, redo : -3-(-1+i) = -2 -i

anonymous · Answer

Oh yeah, thanks @OOOPS.

@zzrock3r: it's a method to eliminate equations out of the matrix. When you multiply the first row with (2+i) and subtract it from the second row you normally can eliminate the second row.

anonymous · Answer

Yup that solves it.

zzr0ck3r · Answer

yes but then we say ots row equivalent, = does not make sense.

zzr0ck3r · Answer

its*

zzr0ck3r · Answer

$\left[\begin{matrix}2-i & -1\ 5 & 4-i\end{matrix}\right]\stackrel{rref}{\iff}\left[\begin{matrix}2-i & -1 \ 0 & 6\end{matrix}\right]$

zzr0ck3r · Answer

something like that

anonymous · Answer

yeah. That's what I did. My problem was just that I couldn't eliminate the second row and got equations for which the eigenvector given in the course didn't work. But I made a mistake, which I couldn't find (I tried this exercise a while ago and had made the same mistake then too).

zzr0ck3r · Answer

my only point is DONT write an equal sign unless the matrices are EXACTLY the same, else use an arrow or something like that.

anonymous · Answer

I know and understand your point. Thank you.

zzr0ck3r · Answer

:)

anonymous · Answer

oH!! Notation warning!! hahahaha  should state out in simple way @zzr0ck3r