How to interpret coordinates of stochastic matrix here?

Question

I have just learnt about Markov chains and have been unsure of where in the matrix $P(0,0)$ refers to exactly (Whether its the top left or bottom right)- I happen to be ignorant of linear algebra, but have researched this topic to no avail, finding notions of coordinates with respect to vectors, which aren't mentioned here. I would appreciate help on how to interpret the coordinates in the below matrix, and in general how to interpret coordinates with no mention of vectors:

We define the transition probability matrix $P$ by $P(0,0)=1-a, P(0,1)=a, P(1,0)=a, P(1,1)=1-a$. That is $$ P=\left[\begin{array}{cc} 1-a & a \\\\ a & 1-a \end{array}\right] . $$Hence, $$ \operatorname{Pr}\left[X_{n+1}=j \mid X_n=i, X_{n-1}, \ldots, X_0\right]=P(i, j), \text { for } n \geq 0 \text { and } i, j \in\{0,1\} $$

thank you

Answer 1

For stochastic matrices, you have to watch the convention, whether the source means row stochastic or column stochastic (that is, which direction sums to 1. This matrix is doubly stochastic so it isn't a big deal). But in general, when we mention a matrix, we use the "rows-then-columns" convention.

The matrix $A$ is $m \times n$, which means it has $m$ rows and $n$ columns. The entry $a_{ij}$ is the entry of $A$ in row $i$, column $j$, always starting from the top left.

In this case, the transition matrix is symmetric, so $P(i, j) = P(j, i)$.

Answer 2

We interpret $P(i,j)$ as the probability to reach state $j$ in step $n+1$ given that you are at state $i$ in step $n$ where $i$ stands for the row (top row is index $0$) and the $j$ stands for the column (the very left column has index $0$). The special thing of probability matrices in Markov chains is that the values in the same row (most commonly, but sometimes you see it for the values in the same column) sum up to 1. So here is an example:

\begin{equation*} P = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \end{equation*}

We require that $a + b = 1 = c + d$ with all values $\geq 0$, and we can determine the probabilities $\text{P}(0,0) = a, \text{P}(0,1) = b, \text{P}(1,0) = c, \text{Pr}(1,1) = d$ to get into state $j$ starting in state $i$.