One way to compare the transition matrices of different subpopulations in a Markov chain model is to compute the Frobenius norm of the difference between the matrices. The Frobenius norm is a measure of the distance between two matrices, and it can be used to quantify the difference between the transition matrices of different subpopulations.
The Frobenius norm, sometimes also called the Euclidean norm (a term unfortunately also used for the vector $L^2$-norm), is matrix norm of an $m \times n$ matrix $\mathbf{A}$ defined as the square root of the sum of the absolute squares of its elements,
$$
\|\mathrm{A}\|_F \equiv \sqrt{\sum_{i=1}^m \sum_{j=1}^n\left|a_{i j}\right|^2}
$$
(Golub and van Loan 1996, p. 55).
To compute the Frobenius norm of the difference between two transition matrices, you first need to compute the difference between the matrices, which can be done by subtracting one matrix from the other. Then, you can compute the Frobenius norm of the difference matrix by taking the square root of the sum of the squares of the elements of the difference matrix.
Once you have computed the Frobenius norm of the difference between the matrices, you can compare this value to a threshold to determine whether the difference is statistically significant. If the norm is larger than the threshold, you can conclude that the transition matrices of the two subpopulations are significantly different from each other.
In terms of interpretation, the Frobenius norm provides a measure of the overall difference between the transition matrices of the two subpopulations. It can be useful for identifying subpopulations that have significantly different state transition patterns from the general population. However, it may be difficult to extract more detailed insights or stylized facts from this measure.
Another approach you might consider is to perform a spectral/eigendecomposition of the transition matrices, which can provide more detailed information about the differences between the matrices. This approach can be more interpretable and might allow you to extract more specific insights about the differences between the subpopulations. However, it may require more advanced mathematical and statistical skills to implement and interpret.