So I am working on eigenvalue problems at the moment, and found something interesting (to me) which I hadn't considered before. Let's have a matrix $A\in \mathbb{R}^{d\times d}$ with eigenvalue decomposition $$A=Q\Lambda Q^{-1}$$ Now consider the eigendecomposition $$Q = Q_2\Lambda_2 Q_2^{-1}.$$ We can keep doing this iteratively and construct a sequence of $\{Q_n\}_{n\in\mathbb{N}}$, where $Q_{n+1}$ is found by $$Q_n = Q_{n+1}\Lambda_{n+1}Q_{n+1}^{-1}.$$ I did some small tests on matrices by hand, and it seems that some initial $A$ will result in the identity matrix for $Q$, but it seems that $A$ being SPD is not a strong enough criterium.
Since it seems that some of these sequences give very familiar results, some the identity some rotations in the (complex) plane, I was wondering if this is a well known property of this iteration. I was wondering if somebody knows more about this or a reference to a book/paper. What I'm mostly interested in is that if $A$ is given, we can maybe already say something about $\lim_{n\rightarrow\infty } Q_n$, and if there are maybe some applications in physics or engineering of this problem.
Any help would be appreciated!