Matrices whose elements are matrices

Question

I've worked with matrices whose elements are objects in a field, such that real numbers, complex numbers, inclusive functions in space of functions, but... Today I was talking to a friend and he asked me about something he saw in his PhD in informatic science that was about "matrices with matrices in their entries" and I know that we can make an arrange of the blocks of the matrices in the entries to form a matrix in a $nxn$ space, for some $n$... but... what use or there is any example of how is useful a matrix with this characteristic?

Answer 1

There are plenty of applications of matrices consisting of "sub" matrices (or as you call "block" matrices). One nice example that comes to mind is the use of block matrices in Absorbing Markov processes. In such a matrix, there are typically four "block" matrices: An Identity matrix, a Zero matrix, a matrix indicating the "flow" from the non absorbing states to the absorbing states and a matrix indicating the flow between non absorbing states. For the future (time going to infinity), the matrix indicating the flow from non absorbing to absorbing is very much of interest as that gives information about probabilities ending up in some absorbing state depending on where you are to begin with. This block matrix is also important for expectation. In order to arrive at such a result, some basic matrix algebra involving block matrices is needed and thus block matrices become important. I will spare you the algebra, but here is an example: Exercise 1.3.2 of Norris, "Markov Chains"

Answer 2

As an example, suppose $A_1,\ldots, A_n$ and $B_1,\ldots, B_m$ are vector spaces. For any linear transformation $f:\bigoplus_{i=1}^n A_i\to\bigoplus_{j=1}^m B_j$ define $f_{ji}=\pi_j\circ f\circ\iota_i$, where $\pi_j:\bigoplus_{j=1}^m B_j\to B_j$ is the canonical projection and $\iota_i:A_i\to\bigoplus_{i=1}^n A_i$ is the canonical injection. Then if we fix bases of all the $A_i$'s and $B_j$'s, the transformation $f$ is uniquely represented by an $m\times n$ matrix whose $(j,i)$-th entry is the matrix representing $f_{ji}$.