$P_{\pi}$ is a irreducible and aperiodic Markov Transition matrix, consider its power $P_{\pi}^{k}$. When $k\rightarrow\infty$, how to prove that the limit of the power of this matrix has the same row? Written as the math equation is $$\lim_\limits{k\rightarrow \infty}P_{\pi}^{k}=\mathbf{1}_{n}d_{\pi}^{T}$$ $\mathbf{1}_{n}=[1 \cdots 1]^{T}$, $d_{\pi}^{T}=[\pi_{1}\cdots \pi_{n}]$ is the eigenvector corresponding to the eigenvalue of 1 of $P_{\pi}$ satisfying $\sum\pi_{i}=1$, and it is also the stationary distribution probability.
I tried diagnolizating the matrix, writing the original matrix as $P_{\pi}=X\Lambda X^{-1}$, then $P_{\pi}^{k}=X\Lambda^{k}X^{-1}$, but I don't know how to handle with the inverse $X^{-1}$...Please help me.