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Section 3.3 Matrix Multiplication

Matrices can also be multiplied together, but this operation is somewhat complicated. Watch the progression in the examples below; basically, the elements of the row of the first matrix are multiplied by the corresponding elements of the column of the second matrix. Matrix multiplication can be written in terms of components as

\begin{equation} C_{ij}=\sum_k A_{ik}B_{kj}\text{.}\tag{3.3.1} \end{equation}

The simplest example is

\begin{equation} \begin{pmatrix} a\amp b \end{pmatrix} \begin{pmatrix} e\\ g \end{pmatrix} = ae+bg\tag{3.3.2} \end{equation}

which should remind you of the dot product. Somewhat more complicated examples are

\begin{equation} \begin{pmatrix} a\amp b\\ c\amp d \end{pmatrix} \begin{pmatrix} e\\ g \end{pmatrix} = \begin{pmatrix} ae+bg\\ ce+dg \end{pmatrix}\tag{3.3.3} \end{equation}


\begin{equation} \begin{pmatrix} a\amp b \end{pmatrix} \begin{pmatrix} e\amp f\\ g\amp h \end{pmatrix} = \begin{pmatrix} ae+bg\amp af+bh \end{pmatrix}\text{.}\tag{3.3.4} \end{equation}

A more general example, combining these ideas, is given by

\begin{equation} \begin{pmatrix} a\amp b\\ c\amp d \end{pmatrix} \begin{pmatrix} e\amp f\\ g\amp h \end{pmatrix} = \begin{pmatrix} ae+bg\amp af+bh\\ ce+dg\amp cf+dh \end{pmatrix}\text{.}\tag{3.3.5} \end{equation}

and a numerical example is

\begin{equation} \begin{pmatrix} 1\amp 2\\ 3\amp 4 \end{pmatrix} \begin{pmatrix} 5\amp 6\\ 7\amp 8 \end{pmatrix} = \begin{pmatrix} 1(5)+2(7)\amp 1(6)+2(8)\\ 3(5)+4(7)\amp 3(6)+4(8) \end{pmatrix} = \begin{pmatrix} 19\amp 22\\ 43\amp 50 \end{pmatrix}\text{.}\tag{3.3.6} \end{equation}

Note however that

\begin{equation} \begin{pmatrix} a\amp b \end{pmatrix} \begin{pmatrix} c\amp d \end{pmatrix}\tag{3.3.7} \end{equation}

is undefined. For matrix multiplication to be defined, the number of columns of the matrix on the left must equal the number of rows of the matrix on the right.


\begin{equation} \begin{pmatrix} 1\amp i\\ -i\amp 1 \end{pmatrix} \begin{pmatrix} 1\\ -1 \end{pmatrix}\text{.}\tag{3.3.8} \end{equation}
\begin{align} \begin{pmatrix} 1\amp i\\ -i\amp 1 \end{pmatrix} \begin{pmatrix} 1\\ -1 \end{pmatrix} \amp=\begin{pmatrix} (1)(1)+(i)(-1)\\ (-i)(1)+(1)(-1) \end{pmatrix}\notag\\ \amp=\begin{pmatrix} 1-i\\ -1-i \end{pmatrix}\text{.}\notag \end{align}

Note: In the special case of a column times a row, the matrix multiplication is called an outer product. The outer product has an important geometric interpretation, especially when the column and row are Hermitian adjoints of each other, see Section 5.7. For example, the following is an outer product:

\begin{equation} \begin{pmatrix} a\\ b \end{pmatrix} \begin{pmatrix} a^{*}\amp b^{*} \end{pmatrix} = \begin{pmatrix} aa^{*}\amp ab^{*}\\ ba^{*}\amp bb^{*} \end{pmatrix}\text{.}\tag{3.3.9} \end{equation}