Originally Posted by

**dwsmith** $\displaystyle \begin{vmatrix}

1-\lambda & 1\\

-2 & 3-\lambda

\end{vmatrix}$

$\displaystyle \lambda=2\pm \mathbf{i}$

When $\displaystyle \lambda=2-\mathbf{i}$, $\displaystyle \begin{bmatrix}

-1+\mathbf{i} & 1\\

-2 & 1+\mathbf{i}

\end{bmatrix}\Rightarrow\begin{bmatrix}

1 & \frac{-1}{2}-\frac{1}{2}\mathbf{i}\\

0 & 0

\end{bmatrix}$.

$\displaystyle x_1=\bigg(\frac{1}{2}-\frac{1}{2}\mathbf{i}\bigg)x_2$

$\displaystyle x_2$

Here is where you went wrong: it must be $\displaystyle x_1=\left(\frac{1}{2}+\frac{1}{2}\,i\right)x_2\iff (1-i)x_1=x_2$ , and thus an eigenvector corresponding to the above eigenvalue is, for example, $\displaystyle \binom{1+i}{2}$

Tonio

$\displaystyle x_2*\begin{bmatrix}

1-\mathbf{i}\\

2

\end{bmatrix}$

What the above and the below mean, anyway??

The book answer is:

$\displaystyle x_2*\begin{bmatrix}

1-\mathbf{i}\\

1

\end{bmatrix}$

Why am I having a discrepancy?