Question

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

Answer 1

Let λ be an eigen value of T

T such that T(x,y) = λ(x,y)

Then we have T(x,y) = λ(x,y)

T(x,y) = λ(x,y) 

So we have

(−3y,x) = (λx,λy)

Thus 

λx = −3y

λx = −3y and x = λy

So we have 

λ(λy) = −3y

Which implies we have λ² = - 3

which has no real solution. So there aren't any real eigen values but has imaginary eigen values.

$\lambda$ ² = -3,

$\lambda$ = $i\sqrt{3}, \, -i\sqrt{3}$ .