Name of the property that certain matrix powers approach the identity matrix

Question

For a proof I'm looking for the name (or proof) of the property that: $$\lim_{t\to\infty}(BAB^{-1})^t = I, \left(\text{ where in my case: } A = \begin{bmatrix}I_n & 0\\0&-I_m \end{bmatrix}\right)$$

With $A,B,I\in\mathbb{R}^{(n+m)\times(n+m)}$

Answer 1

The only matrix $A$ that satisfies the given condition is the identity matrix.

Consider the Jordan form $J$ of $A$ over $\mathbb C$. The given condition implies that $\lim_{k\to\infty}J^k=I$, we have $\lim_{k\to\infty}\lambda^k=1$ for every eigenvalue $\lambda$. Hence all eigenvalues of $A$ or $J$ are equal to $1$. It follows that $J$ has no nontrivial Jordan block, otherwise $J^k$ would possess a super-diagonal entry whose value is $k$, contradicting the fact that $J^k$ converges to $I$. Thus $J=I$, i.e. $A=I$.