Block (partitioned) matrices

The gist of it is that you can take any matrix and "block" it into certain forms. For example, you can have

The really cool thing is that you can do matrix multiplication on these sub-blocks as if they were individual elements!

This can be proven by simply doing them out.

Block matrix: $\begin{bmatrix} A & B \\C&D\end{bmatrix}^T = \begin{bmatrix} A^T & C^T \\B^T&D^T\end{bmatrix}$ . Note how the transpose happens to the block matrix and it propagates inside!

Block Matrices multiplication

You can take your matrix $A$ and chop it into a "block matrix", where each element is a matrix. The rules of matrix multiplication still apply! You can treat the blocks as numbers, and all the products you compute are matrix products of the block elements. This is a result of the profound linearity of linear maps