Answer to Question #213774 in Linear Algebra for Simphiwe Dlamini

Question

Suppose T $\isin$ L(R²) is dened by T(x,y) = (-3y,x). Find the eigenvalues of T.

Answer 1

Given that T is a linear transformation

T(x,y)=(-3y,x)

we know that the basis of R² is {(1,0),(0,1)}

then,

T(1,0)=(-3(0),1)

T(1,0)=(0,1)

T(1,0)=0(1,0)+1(0,1)...........(1)

And,

T(0,1)=((-3)1,0)=(-3,0)

(0,1)=-3(1,0)+0(0,1)............(2)

Now the matrix representation of given linear transformation with respect to standard basis is given as,

[T]=$\begin{bmatrix} 0 & -3 \\ 1 & 0 \end{bmatrix}$

to find the eigen values of T we need to find the eigen values of matrix [T]

let A=[T]

then, A=$\begin{bmatrix} 0 & -3 \\ 1 & 0 \end{bmatrix}$

for eigen values put |A-$\lambda$ I|=0

where$\lambda$ is eigen value of A

then,

$\begin{bmatrix} 0 & -3 \\ 1 & 0 \end{bmatrix}$ -$\begin{bmatrix} \lambda & 0 \\ 0 & \lambda \end{bmatrix}$ =A-$\lambda$ I

A-$\lambda$ I=$\begin{bmatrix} -\lambda & -3 \\ 1 & -\lambda \end{bmatrix}$

put A-$\lambda$ I=0

then,

$\begin{bmatrix} -\lambda & -3 \\ 1 & -\lambda \end{bmatrix}$ =0

$\lambda^2$ +3=0

$\lambda^2$ =-3

$\lambda=\sqrt3i,-\sqrt3i$

$\therefore$ The eigen values of T are $\sqrt3i,-\sqrt3i$