In Grinstead & Snell (2006), on page 417, the authors discuss how to find the 'fundamental matrix' of an absorbing Markov chain. They define two matrices:
-
The matrix $\textbf{I}$ is "an r-by-r identity matrix", where r is the number of absorbing states.
-
The matrix $\textbf{Q}$ as "a t-by-t matrix", where t is the number of transient states. $\textbf{Q}$ is the matrix of transition probabilities between each of the transient states.
Later on (p.419), they calculate the matrix $\textbf{I-Q}$. Apologies if I am showing my naivety, but how can you subtract matrices of different sizes?
On the relevant Wikipedia page, the same operation is discussed but $\textbf{I}$ is defined not as the identity matrix for the absorbing states but rather as the identity matrix for the transient states. Now the two matrices $\textbf{I}$ and $\textbf{Q}$ are the same size and we can subtract one from the other.
Is this a simple typo in Grinstead & Snell (2006), or am I being dim? (i.e. when $\textbf{I}$ is introduced, should it just be defined as a t-by-t identity matrix, rather than an r-by-r identity matrix?)
Best Answer
$\mathbf I$ can represent an identity matrix of any size depending on the context, so $\mathbf I$ in the expression of $\mathbf I-\mathbf Q$ should be interpreted as the $t\times t$ identity matrix to match the size of $\mathbf Q$.