Building block matrices using the tensor product

Block matrices appear very frequently in science and engineering problems, especially numerical solutions of differential equations. They are matrices which are made of blocks that possess a certain structure and properties, arranged in a certain way. For example, the matrix

has a repeating pattern of 2x2 matrices across the main diagonal. If we set

Then we can write

This is nice and all, but how can we start from A and 0 and then construct this new matrix using some kind of mathematical operation? It turns out, there is a very straight-forward way to do so, and it is called the matrix tensor product or Kronecker product. It works like this:

where M is a matrix of any size. The operation works as follows: for the matrix on the left of the tensor product symbol, take each element and multiply it by the entire matrix on the right, in this case M and this then becomes a “block element” of the new matrix, in that same relative position. This operation is…