Order of x and y-components for eigenvectors

Question

I am confused when it comes to writing the eigenvectors of a matrix. The problem that I am having specifically is determining whether or not I set x = 1 or y = 1 for the eigenvector of a 2x2 matrix. For example, the 2x2 matrix below,

$$ \begin{matrix} 3/4 & \sqrt{3}/4 \\ \sqrt{3}/4 & 1/4 \\ \end{matrix} $$

The eigenvalues are 0 and 1, so I get the following equations to determine the x and y-component of the eignenvectors:

$$ 3x + \sqrt{3}y = 0 $$ $$ 3x + \sqrt{3}y = 4 $$

What I did in both cases was made x the subject of y and set x = 1 to get the value of y, but in the solution to this question y was set to 1 instead. So the order of my eigenvector components was wrong. How do I know when to set x equal to or y equal to 1? Thank you

Answer 1

The eigenvalue $\lambda = 0$ gives the equation $$\left[ \begin{matrix} \frac{3}{4} & \frac{\sqrt{3}}{4} \\ \frac{\sqrt{3}}{4} & \frac{1}{4} \\ \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ \end{matrix} \right] = 0 \left[ \begin{matrix} x \\ y \\ \end{matrix} \right]$$ This gives $y=-\sqrt{3}x$. Any matrix $v=\left[ \begin{matrix} x \\ y \\ \end{matrix}\right]$ satisfying this equation is an eigenvector for the eigenvalue $0$ (except of course $x=0$). Put $\lambda = 1$ in place of the $\lambda = 0$ and repeat.