<s>[INST] Complex eigenvectors... I can't get the right answer even though I'm using software I'm using software to calculate my eigenvectors, and I can't get the correct answers...
I have $$A = \begin{pmatrix} 1 & 2+i \\ 2+i & 1 \end{pmatrix}.$$ I solved the eigenvalues to be $3+i, -1-i$ (which is correct according to my software). Then I compute $\ker(A - (3+i)I)$ and $\ker(A-(-1-i)I)$, which gives me eigenvectors of $(1,)^T, (-1,0)^T$. But, plugging them in, they don't satisfy the requirement of an eigenvector. Can someone show me how my method is wrong and how to arrive at the correct answer?
 [/INST] We all agree that given
$$
  \mathbf{A} = \left[
\begin{array}{cc}
 1 & 2+i \\
 2+i & 1 \\
\end{array}
\right]
$$
the eigenvalues are
$$
 \lambda \left( \mathbf{A} \right) = \left\{ 3 + i, -1 - i \right\}
$$
The eigenvectors are
$$
  v_{1} = 
\color{blue}{\left[
\begin{array}{c}
 1 \\
 1 \\
\end{array}
\right]}, \qquad 
  v_{2} = 
\color{red}{\left[
\begin{array}{r}
-1 \\
 1 \\
\end{array}
\right]}
$$
The eigenvector equations are
$$  
\begin{align}
%
  \left( \mathbf{A} - \lambda_{1} \mathbf{I}_{2} \right) \cdot v_{1} 
&= \mathbf{0} \\[5pt]
%
\left[
\begin{array}{rr}
 -2-i & 2+i \\
 2+i & -2-i \\
\end{array}
\right] 
\color{blue}{\left[
\begin{array}{r}
 1 \\
 1 \\
\end{array}
\right]}
&=
\left[
\begin{array}{r}
 0 \\
 0 \\
\end{array}
\right] \\[10pt]
\end{align}
$$
$$  
\begin{align}
  \left( \mathbf{A} - \lambda_{2} \mathbf{I}_{2} \right) \cdot v_{2} 
&= \mathbf{0} \\[5pt]
%
\left[
\begin{array}{rr}
 2+i & 2+i \\
 2+i & 2+i \\
\end{array}
\right] 
\color{red}{\left[
\begin{array}{r}
-1 \\
 1 \\
\end{array}
\right]}
&=
\left[
\begin{array}{r}
 0 \\
 0 \\
\end{array}
\right] \\[10pt]
%
\end{align}
$$

We don't know what software package the OP used. So to advance the discussion, here is Mathematica output.



Of course we can solve the eigenvalue equation directly.
$$
 \mathbf{A} v = \lambda v
$$
The answer is the same.


</s>