Question

Suppose T in End(F^2) is defined by
T(w,z)=(-z,w)
Find the eigenvalues and eigenvectors of T if F=R.

Answer 1

The standard basis for R² is (1,0) and (0,1)

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

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

Thematrix of the given linear transformation is given by

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

The characteristic polynomial is given by

$Det(A-xI) = \begin{vmatrix} -x& -1\\ 1& -x \end{vmatrix}$ =0

$x^2 +1 =0\\$

But if F=R, then the characteristic polynomial has no zeroes in R.

So, there are no eigenvalues and no eigenvectors when F=R.