Matrix multiplication

How do you multiply matrices? I realized that the technique that I learned back in the days is not as common as I thought. I often see my students struggling to perform simple matrix multiplications, and I would argue that part of the reason is that they write them side-by-side. This is also what I see in the books I could get my hands on, and in some popular videos online.

To perform the multiplication $AB$ of two matrices, I learned to write things in the way depicted in the picture below.

The advantages are multiple. First, to compute the element in position $(i,j)$ of $AB$ , the definition of matrix multiplication says that we need the $i$ -th row of $A$ and the $j$ -th column of $B$ , which we can neatly see at the left and the top of position $(i,j)$ .

Therefore, doing the operation $A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + \cdots$ requires only a minimum of mental gymnastics and eye movement.

Multiply the term connected by the dotted lines, then add all of them.

Added advantage:the matrix multiplication formula

$\displaystyle (AB)_{i,j} = \sum_{k=1}^{n} A_{i,k} B_{k,j}$ ,

is right there under your eyes, no need to remember it. Provided of course that we remember that an element $(i,j)$ is in the $i$ -th row and $j$ -th column, not vice versa, but this is convention and we cannot do anything about it.

A third selling point of this method is that it makes it easy to see the size of matrices.

We can multiply $A$ and $B$ if we can “lay down” B on A after rotating it $90^{\circ}$ . In other words, the top part of $A$ and left part of $B$ can be glued together perfectly. In symbols, we can multiply a $m \times n$ matrix by a $n \times p$ matrix.

The matrix multiplication $AB$ fits snuggly in the space left on the right of $A$ and the bottom of $B$ . In other words, the product of a $m \times n$ matrix by a $n \times p$ matrix is a $m \times p$ matrix.

As far as I am concerned, this is the only way I can recover all these formulas.

Consider matrices $A_1, \dots, A_k$ of size respectively $(m_1,n_1), \dots, (m_k,n_k)$ . Under what conditions is the matrix product $A_1 \cdots A_k$ well-defined? What is the size of the resulting matrix?

Consider a square $n \times n$ matrix $A$ , and the column vector $\mathbf{1}$ of size $n \times 1$ that is filled with $1$ . Explain in words what $A \mathbf{1}$ , $\mathbf{1}^T A$ and $\mathbf{1}^T A \mathbf{1}$ are. The exponent ${}^T$ is the transpose.