# Do matrix multiplication rules apply when multiplying matrices made up of smaller matrices?

I came across a proof that did the following.

$$a$$, $$b$$, $$c$$ are $$3 \times 1$$ vectors. $$A$$ is a $$3 \times 3$$ matrix and $$d$$ is a $$3 \times 1$$ vector

$$\begin{bmatrix}A & d\end{bmatrix} \begin{bmatrix} a & b & c \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} Aa & Ab & Ac +d\end{bmatrix}$$

So we are left multiplying a $$4 \times 3$$ matrix by a $$3 \times 4$$ matrix, but the elements of both matrices are themselves vectors and matrices.

I know this is correct because the rest of the proof in the paper follows. But, I don't really know how to justify using regular matrix multiplication properties to multiply two matrices made up of matrices. For example, $$Aa$$ is a matrix-vector product, whereas in regular matrix multiplication it would be some scalar product.

1. Is there a name for this kind of linear algebra property & are there more like it?

2. Why does it work? (Just a reference would suffice)

• You can try reading about block matrices and their arithmetic operations. Dec 17 '18 at 10:31
• en.wikipedia.org/wiki/Block_matrix
– user856
Dec 17 '18 at 10:33
• Thank you both! I didn't know there was something called Block Matrix. Dec 17 '18 at 10:34
• I'm fairly sure that matrix multiplication is thoroughly recursive in that respect. Don't rely on what I've said though ... I just have a recollection of learning that it is. Dec 17 '18 at 11:00

This block matrix notation is very useful. The meaning of the notation is clear: for example, if $$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}, \quad B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}, \quad C = \begin{bmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{bmatrix}, \quad D = \begin{bmatrix} d_{11} & d_{12} \\ d_{21} & d_{22} \end{bmatrix}$$ then $$\begin{bmatrix} A & B \\ C & D \end{bmatrix}$$ denotes the $$4 \times 4$$ matrix $$\begin{bmatrix} A & B \\ C & D \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & b_{11} & b_{12} \\ a_{21} & a_{22} & b_{21} & b_{22} \\ c_{11} & c_{12} & d_{11} & d_{12} \\ c_{21} & c_{22} & d_{21} & d_{22} \end{bmatrix}.$$ It is straightforward to prove the following basic rules for matrix multiplication using block notation:
• $$A \begin{bmatrix} B & C \end{bmatrix} = \begin{bmatrix} AB & AC \end{bmatrix}$$.
• $$\begin{bmatrix} A \\ B \end{bmatrix} C = \begin{bmatrix} AC \\ BC \end{bmatrix}.$$
• $$\begin{bmatrix} A & B \end{bmatrix} \begin{bmatrix} C \\ D \end{bmatrix} = AC + BD.$$
(For each rule, we must assume that the matrices $$A, B, C$$, and $$D$$ have compatible shapes.)
\begin{align} \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} X \\ Y \end{bmatrix} &= \begin{bmatrix} \begin{bmatrix} A & B \end{bmatrix} \begin{bmatrix} X \\ Y \end{bmatrix} \\ \begin{bmatrix} C & D \end{bmatrix} \begin{bmatrix} X \\ Y \end{bmatrix} \end{bmatrix} \\ &= \begin{bmatrix} AX + BY \\ CX + DY \end{bmatrix} \end{align} (assuming that the matrices $$A, B, C, D, X$$, and $$Y$$ have compatible shapes).