<s>[INST] any physical interpretation for these two matrices? I know that the physical interpretation of $\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin\theta & \cos \theta \end{pmatrix}$ is the rotation matrix.
But what are the physical interpretations of the matrices $\begin{pmatrix} \sin \theta & \cos \theta \\ -\cos\theta & \sin \theta \end{pmatrix}$ and $\begin{pmatrix} \sec \theta & \tan \theta \\ \tan\theta & \sec \theta \end{pmatrix}$ ?
I have tried with a couple of values but was unable to arrive at a conclusion.
 [/INST] $$\begin{pmatrix} \sin \theta & \cos \theta \\ -\cos\theta & \sin \theta \end{pmatrix} = \begin{pmatrix} \cos \left(\theta-\frac{\pi}{2}\right) & -\sin \left(\theta-\frac{\pi}{2}\right) \\ \sin \left(\theta-\frac{\pi}{2}\right) & \cos \left(\theta-\frac{\pi}{2}\right) \end{pmatrix}$$
So it is also a rotation matrix.
A physical interpretation for the other is not so clear to me. But the diagonalization may help.
$$\begin{pmatrix} \sec \theta & \tan \theta \\ \tan\theta & \sec \theta \end{pmatrix}=\begin{pmatrix} -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}  \\ \frac{1}{\sqrt{2}}  & \frac{1}{\sqrt{2}} \end{pmatrix} \begin{pmatrix} \sec \theta - \tan \theta & 0  \\ 0  & \sec \theta + \tan \theta \end{pmatrix}\begin{pmatrix} -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}  \\ \frac{1}{\sqrt{2}}  & \frac{1}{\sqrt{2}} \end{pmatrix}
$$
</s>