How can I find eigen vectors of this matrix analytically?

Question

Matrix $A$ is given by;

$A = \begin{bmatrix} 0 & 1 \\ -4 & 0 \end{bmatrix}$

First I find the eigen values as $\lambda_1$ and $\lambda_2$ as $2i$ and $-2i$. And then to find the eigen vector of the eigen value $2i$, I do;

$\begin{bmatrix}2i & -1\\4&2i\end{bmatrix}\cdot\begin{bmatrix}w_1\\w_2\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix}$

from this point equations look like;

$\begin{align*}2iw_1 - w_2 &= 0 \\[0.4em] 4w_1+2iw_2 &= 0\end{align*}$

and I cannot do the elimination at this step because when I try to cancel one variable, the other one is cancelled out too..

Answer 1

Basically, the two equations are the same and you get from them, $$w_1=\frac{w_2}{2i}$$ So one eigenvector corresponding to the eigenvalue $2i$ is $[1\quad 2i]^T$ because it satisfies the equations. Remember, there can be infinitely many eigenvectors corresponding to a single eigenvalue.

Answer 2

You are supposed to get an under-determinate system. Eigenvectors are defined up to a constant. Add one more equation, say $w_1 + w_2 = 1$