I'm reading this book about quantum mechanics. The author wants the reader to do the following exercise:
Find the eigenvectors and eigenvalues of the operator $$A = \pmatrix{\cos \theta & \sin \theta \\ \sin \theta & -\cos \theta} .$$
He says to use the following trial eigenvector and find $\alpha$ in terms of $\theta$:
$$|V\rangle = \pmatrix{\cos \alpha \\ \sin \alpha} .$$
I found the eigenvalues: $λ_1=+1$ and $λ_2=-1$ for the spin, but I can't seem to find the eigenvectors in the book:
$[ \cos(θ/2) ; \sin(θ/2) ]$ for $λ_1=+1$
$[ -\sin(θ/2) ; \cos(θ/2) ]$ for $λ_2=-1$
I get completely different eigenvectors, which work but the ones above work too. He then uses those vectors to find the probability of getting $+1$ state:
$P(+1) = ([1 0]\times[ \cos(θ/2) ; \sin(θ/2) ])^2$ $P(+1) = (\cos(θ/2))^2$
any help for finding these eigenvectors?
EDIT: I get the following eigenvectors:
$[ \cotθ-1 ; 1 ]$ for $λ=-1$ $[ \cotθ+1 ; 1 ]$ for $λ=+1$
As you know if $$Av=\lambda v,$$ then also for $w=\alpha v$ you'll get $$Aw=\alpha Av=\alpha\lambda v=\lambda w$$
Particularly, for your case, one of eigenvalues may look like $$\begin{pmatrix} \cot t+\csc t\\1 \end{pmatrix} $$then $$ \sin\frac{t}{2} \begin{pmatrix} \cot t+\csc t\\1 \end{pmatrix} =\begin{pmatrix} \cos\frac{t}{2} \\\sin\frac{t}{2} \end{pmatrix}$$
With the second it is very similar.