I am currently doing a question to get all possible eigenvectors of a matrix, and while I believe I got the correct answer (and WolframAlpha also says that my answer is correct), I was marked wrong. I just want to know if I made some mistake, or perhaps the answer that the online system I had was wrong. The question is as follows:
Find all distinct (real or complex) eigenvalues of A. Then find the basic eigenvectors of A corresponding to each eigenvalue.
For each eigenvalue, specify the number of basic eigenvectors corresponding to that eigenvalue, then enter the eigenvalue followed by the basic eigenvectors corresponding to that eigenvalue.
$$
A =
\begin{bmatrix}
-6 & 5 \\
-10 & 8
\end{bmatrix}
$$
I was able to determine the eigenvalues to be $1+i$ as well as $1-i$, which it did accept, except I got (as well as WolframAlpha) the Eigenvectors below:
$$
1+i:
\begin{bmatrix}
\frac{7}{10}+\frac{i}{10} \\
1
\end{bmatrix}
$$
$$
1-i:
\begin{bmatrix}
\frac{7}{10}-\frac{i}{10} \\
1
\end{bmatrix}
$$
While the correct answer it gave was as follows:
$$
1+i:
\begin{bmatrix}
-2+i \\
-3+i
\end{bmatrix}
$$
$$
1-i:
\begin{bmatrix}
-2-i\\
-3-i
\end{bmatrix}
$$
Would anyone be able to tell me if I am correct and there is some issue with my homework system, or I am wrong and I should be quadruple checking my work? This has been driving me crazy all night.
Best Answer
Hint:
$$ \left(\frac{7}{10}+\frac{i}{10}\right)(-3-i)=-2-i $$ do the same for the other eigenvector. $$ \left(\frac{7}{10}-\frac{i}{10}\right)(-3+i)=-2+i $$
It seems that you have changed the eigenvectors.( see http://www.wolframalpha.com/input/?i=eigenvalues+%7B%7B-6%2C5%7D%2C%7B-10%2C8%7D%7D)