<s>[INST] block matrix multiplication If $A,B$ are $2 \times 2$ matrices of real or complex numbers, then
$$AB = \left[
\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right]\cdot
\left[
\begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array} \right]
= 
\left[
\begin{array}{cc} a_{11}b_{11}+a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22} \\ a_{21}b_{11}+a_{22}b_{21} & a_{22}b_{12}+a_{22}b_{22} \end{array} \right]
$$
What if the entries $a_{ij}, b_{ij}$ are themselves $2 \times 2$ matrices?  Does matrix multiplication hold in some sort of "block" form ?
$$AB = \left[
\begin{array}{c|c} A_{11} & A_{12} \\\hline A_{21} & A_{22} \end{array} \right]\cdot
\left[
\begin{array}{c|c} B_{11} & B_{12} \\\hline B_{21} & B_{22} \end{array} \right]
= 
\left[
\begin{array}{c|c} A_{11}B_{11}+A_{12}B_{21} & A_{11}B_{12}+A_{12}B_{22} \\\hline A_{21}B_{11}+A_{22}B_{21} & A_{22}B_{12}+A_{22}B_{22} \end{array} \right]
$$
This identity would be very useful in my research.
 [/INST] It depends on how you partition it, not all partitions work.
For example, if you partition these two matrices 
$$\begin{bmatrix}
    a & b & c \\
    d & e & f \\
    g & h & i
  \end{bmatrix},
  \begin{bmatrix}
    a' & b' & c' \\
    d' & e' & f' \\
    g' & h' & i'
  \end{bmatrix} $$ 
in this way 
$$ \left[\begin{array}{c|cc}a&b&c\\  d&e&f\\ \hline g&h&i \end{array}\right],
 \left[\begin{array}{c|cc}a'&b'&c'\\  d'&e'&f'\\ \hline g'&h'&i' \end{array}\right] $$
and then multiply them, it won't work. But this would
$$\left[\begin{array}{c|cc}a&b&c\\ \hline d&e&f\\ g&h&i \end{array}\right] ,\left[\begin{array}{c|cc}a'&b'&c'\\ \hline d'&e'&f'\\ g'&h'&i' \end{array}\right] $$
What's the difference? Well, in the first case, all submatrix products are not defined, like $\begin{bmatrix}
    a \\ d  \\ \end{bmatrix}$ cannot be multiplied with $\begin{bmatrix}
    a' \\ d'  \\ \end{bmatrix}$
So, what is the general rule? (Taken entirely from the Wiki page on Block matrix)
Given, an $(m  \times p)$ matrix $\mathbf{A}$ with $q$ row partitions and $s$ column partitions
$$\begin{bmatrix}
\mathbf{A}_{11} & \mathbf{A}_{12} & \cdots &\mathbf{A}_{1s}\\
\mathbf{A}_{21} & \mathbf{A}_{22} & \cdots &\mathbf{A}_{2s}\\
\vdots          & \vdots          & \ddots &\vdots \\
\mathbf{A}_{q1} & \mathbf{A}_{q2} & \cdots &\mathbf{A}_{qs}\end{bmatrix}$$
and a $(p \times n)$ matrix $\mathbf{B}$ with $s$ row partitions and $r$ column parttions
$$\begin{bmatrix}
\mathbf{B}_{11} & \mathbf{B}_{12} & \cdots &\mathbf{B}_{1r}\\
\mathbf{B}_{21} & \mathbf{B}_{22} & \cdots &\mathbf{B}_{2r}\\
\vdots          & \vdots          & \ddots &\vdots \\
\mathbf{B}_{s1} & \mathbf{B}_{s2} & \cdots &\mathbf{B}_{sr}\end{bmatrix}$$
that are compatible with the partitions of $\mathbf{A}$, the matrix product
$
\mathbf{C}=\mathbf{A}\mathbf{B}
$
can be formed blockwise, yielding $\mathbf{C}$ as an $(m\times n)$ matrix with $q$ row partitions and $r$ column partitions.
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