I just want to check that the matrix below is not positive definite:
$$\begin{bmatrix}\cos(\theta) & -\sin(\theta) \\\sin(\theta) & \cos(\theta)\end{bmatrix}$$
I think that it is not because the upper left 1*1 determinant is just $cos(\theta)$ and this is not positive for every value of $\theta$
Is this correct?
You'll just have to see that this is a rotation and then go back to the definition of definiteness. $R_\theta$ is positively definite if $u\cdot R_\theta u>0$ for all $u\ne 0$. Now we have thhat $u\cdot R_\theta u=|u|^2\cos\theta > 0$ for all $u\ne 0$ if and only if $\cos\theta > 0$.