introduction, motivation …
It turns out that it is relatively simple to implement infinite matrices in the Wofram Language. The implementation we will develop here and in further posts will include:
Since row and columns vectors can be treated as matrices with one row and or one column, these 4 operations will also take vector arguments.
Why do this? The original motivation was to demonstrate the probability transition matrix for the Poisson process. It looks something like this:
\[ P = \left( \begin{array}{cccccc} 1-p & p & 0 & \cdot & \cdot & \cdot \\ 0 & 1-p & p & \cdot & \cdot & \cdot \\ 0 & 0 & 1-p & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \end{array} \right) \]
where \(p\) is the probability of going from state \(i\) to state \(i + 1\) and \(1 - p\) is the probability of going from state \(i\) to state \(i\). The three rows and three columns of dots, as is tradition, symbolize the remaining, infinite, part of the matrix.
What about \(P^{2}\)? This matrix corresponds to transition probabilities after two steps of the Poisson process. With a little head scratching you can come up with the following picture:
\[ P^{2} = \left( \begin{array}{cccccc} (1-p)^2 & 2 (1-p) p & p^2 & \cdot & \cdot & \cdot \\ 0 & (1-p)^2 & 2 (1-p) p & \cdot & \cdot & \cdot \\ 0 & 0 & (1-p)^2 & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \end{array} \right) \]
Simple enough. What about \(P^{15}\)? With some clever usage of functional programming we will persuade Mathematica to produce the answer (actually, it is possible to work out the analytical form of any element in \(P^{n}\) and verify that this result is correct):
\[ P^{15} = \left( \begin{array}{cccccc} (1-p)^{16} & 16 (1-p)^{15} p & 120 (1-p)^{14} p^2 & \cdot & \cdot & \cdot \\ 0 & (1-p)^{16} & 16 (1-p)^{15} p & \cdot & \cdot & \cdot \\ 0 & 0 & (1-p)^{16} & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \end{array} \right) \]
Our Wolfram Language implementation will also allow us to calculate any segment of the infinite matrix. Here are rows \(32 ... 34\) and columns \(35 ... 37\) of \(P^{15}\):
\[ \left( \begin{array}{ccccccccc} \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & 560 (1-\text{pp})^{13} \text{pp}^3 & 1820 (1-\text{pp})^{12} \text{pp}^4 & 4368 (1-\text{pp})^{11} \text{pp}^5 & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & 120 (1-\text{pp})^{14} \text{pp}^2 & 560 (1-\text{pp})^{13} \text{pp}^3 & 1820 (1-\text{pp})^{12} \text{pp}^4 & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & 16 (1-\text{pp})^{15} \text{pp} & 120 (1-\text{pp})^{14} \text{pp}^2 & 560 (1-\text{pp})^{13} \text{pp}^3 & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \end{array} \right) \]
Obviously this is not magic and will not work for all infinite matrices. One assumption that we will make is that any row of a matrix and any column of a matrix has to have a finite number of elements. Even with this caveat our code might be a good tool to get more insight into many physical processes, whose description revolves around infinite matrices.
In the next post we will tackle infinite matrix representation and transposition.