How to show that a matrix is the eigenvector matrix

Question

I have the matrix $$\mathrm{T}=\left[\begin{array}{cc} e^{B+J} & e^{-J} \\ e^{-J} & e^{-B+J} \end{array}\right]$$ With the eigenvalues: $$\lambda_{1,2}=e^{J} \cosh (B) \pm \sqrt{e^{2 J} \sinh ^{2}(B)+e^{-2 J}}, \quad \cot (2 \phi)=e^{2 J} \sinh (B), \quad 0<\phi<\frac{\pi}{2}$$ And I would like to show that $$S=\left(\begin{array}{cc} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{array}\right)$$ Is the matrix of eigenvectors for $T$. What would be the simplest way to show this?