Question

Find the eigenvalues and eigenvectors for trigonometric functions

Answer 1

Problem statement:

Diagonalize the following matrix:

(cosθsinθsinθ−cosθ)(cos⁡θsin⁡θsin⁡θ−cos⁡θ)

My attempt:

Ie found that the eigenvalues of this matrix are λ=−1λ=−1 or λ=1λ=1, so I plugged in λ=−1λ=−1:

(cosθ+1sinθsinθ−cosθ+1)(xy)=0(cos⁡θ+1sin⁡θsin⁡θ−cos⁡θ+1)(xy)=0

I tried to solve this, but I got 2xsinθ=02xsin⁡θ=0. How do I find the eigenvectors?