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I was trying to get intuition on the notion of similarity of two matrices by looking at the set of rotations in $\mathbb{R}^3$ which are similar to a given rotation. My thoughts on this:

Any rotation can be specified by an axis and an angle $(\hat{n}_1,\theta_1)$, with $\theta_1\in [-\pi,\pi)$. If a rotation with different parameters $(\hat{n}_2, \theta_2)$ is given, one can perform a change of basis so that this new rotation, when plotted on the new coordinate axes, has the same axis (and therefore direction of rotation). Then, the rotations are the same when compared across coordinate systems if and only if the angle of rotation is the same. I would guess that two rotations with angles $\theta_1, \theta_2,$ with $ \theta_1+\theta_2=0$ are also equivalent, because the axis could be flipped to the other side to give the same rotation.

So, it seems that two rotations are equivalent if and only if, viewing the rotation from a coordinate system aligned or anti-aligned with the axis, they rotate by the same angle. However, this is all only intuition. Is the conclusion correct? If not, what is the correct conclusion here?

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The conclusion is correct. Instead of switching coordinate systems, you can work with the rotation matrices themselves. Two rotation matrices $\Omega_1$, $\Omega_2$ in three dimensions are similar exactly if the cosines of their angles of rotation are equal (which, if you choose the angles in $[-\pi,\pi)$, is equivalent to the angles being equal or their sum being zero). For the “if” direction, rotate the axis of $\Omega_1$ into that of $\Omega_2$ (or its inverse if necessary), using some rotation matrix $R$, then apply $\Omega_2$, and then apply $R^{-1}$ to rotate the axis back to its original position. The combined effect is that of $\Omega_1$. For the other direction, note that the trace of a rotation matrix in three dimensions is $1+2\cos\theta$, where $\theta$ is the angle of rotation. Since similar matrices have the same trace, rotation matrices with different values of $\cos\theta$ are not similar.

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