I am wondering what $\pi_0P^n(i_n)$ means with the 'component $i_n$'- what in the matrix $\pi_0P^n$ are we accessing with this $(i_n)$? What is the 'component $i_n$'- a certain row of the matrix?
For context, I am ignorant of linear algebra and have just been learning about Markov Chains, but my research online hasn't resolved this issue. Here is the area where I have encountered this notation (the specific part I am asking about is at the bottom):
One defines a general finite Markov chain as follows. The state space is $\mathscr{X}=\{1,2, \ldots, K\}$ for some finite $K$. The transition probability matrix $P$ is a $K \times K$ matrix such that $$ P(i, j) \geq 0, \forall i, j \in \mathscr{X} $$ and $$ \sum_{j=1}^K P(i, j)=1, \forall i \in \mathscr{X}$$The initial distribution is a vector $\pi_0=${$ \pi_0(i), i \in \mathscr{X}$} where $\pi_0(i) \geq 0$ for all $i \in \mathscr{X}$ and $\sum_{i \in \mathscr{X}} \pi_0(i)=1$. One then defines the random sequence {$X_n, n=0,1,2, \ldots$} by $$ \begin{aligned} & \operatorname{Pr}\left[X_0=i\right]=\pi_0(i), i \in \mathscr{X} \\ & \operatorname{Pr}\left[X_{n+1}=j \mid X_n=i, X_{n-1}, \ldots, X_0\right]=P(i, j), \forall n \geq 0, \forall i, j \in \mathscr{X} . \end{aligned}$$Note that $$ \operatorname{Pr}\left[X_0=i_0, X_1=i_1, \ldots, X_n=i_n\right] $$ $$ \begin{aligned} & =\operatorname{Pr}\left[X_0=i_0\right] \operatorname{Pr}\left[X_1=i_1 \mid X_0=i_0\right] \operatorname{Pr}\left[X_2=i_2 \mid X_0=i_0, X_1=i_1\right] \cdots \operatorname{Pr}\left[X_n=i_n \mid X_0=i_0, \ldots, X_{n-1}=i_{n-1}\right] \\ & =\pi_0\left(i_0\right) P\left(i_0, i_1\right) \cdots P\left(i_{n-1}, i_n\right) . \end{aligned}$$Consequently, $$ \begin{aligned} \operatorname{Pr} & {\left[X_n=i_n\right]=\sum_{i_0, \ldots i_{n-1}} \operatorname{Pr}\left[X_0=i_0, X_1=i_1, \ldots, X_n=i_n\right] } \\ & =\sum_{i_0, \ldots i_{n-1}} \pi_0\left(i_0\right) P\left(i_0, i_1\right) \cdots P\left(i_{n-1}, i_n\right) \\ & =\pi_0 P^n\left(i_n\right) \end{aligned} $$ where the last expression is component $i_n$ of the product of the row vector $\pi_0$ times the $n$-th power of the matrix $P$.