Kunze Linear Algebra · Problem 3.4.8

Linear Transformations

Mathematics — Methods · similar-matrices-rotation

Problem

Let

\theta

be a real number. Prove that the following two matrices are similar over the field of complex numbers:

\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}, \quad \begin{bmatrix} e^{i\theta} & 0 \\ 0 & e^{-i\theta} \end{bmatrix}

(Hint: Let

T

be the linear operator on

C^2

which is represented by the first matrix in the standard ordered basis. Then find vectors

\alpha_1

and

\alpha_2

such that

T\alpha_1 = e^{i\theta}\alpha_1

T\alpha_2 = e^{-i\theta}\alpha_2

, and

\{\alpha_1, \alpha_2\}

is a basis.)