Question

Suppose T € L(R^2) is defined by T(x;y) = ((3y; x). Find the eigenvalues of T.

Answer 1

Let us solve the equation $T(x,y)=(\lambda x, \lambda y)$ for $\lambda$ :

$\begin{cases} 3y=\lambda x \\ x=\lambda y \end{cases}$ , now using the substitution we find that $\begin{cases} 3y=\lambda^2y \\ x= \lambda^2x/3 \end{cases}$.

Therefore, the possible solutions for $\lambda$ are $\sqrt{3}, -\sqrt{3}$ (as at least one of numbers $x,y$ is non-zero).

Another way to calculate it could be expressing $T$ in a matrix form :

$T= \begin{pmatrix} 0 & 3 \\ 1 & 0 \end{pmatrix}$

and find the characteristic polynomial :

$\lambda^2-3=0$

which yields the same result.