Matrix Multiplication
Overview
There are two primary definitions of multiplication with matrices.
Multiplication by a Scalar
Mulitplying a matrix by a scalar results in a matrix where every element of the original matrix is just
multiplied by the given scalar.
{%
c \begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33} \\
\end{bmatrix} =
\begin{bmatrix}
ca_{11} & ca_{12} & ca_{13} \\
ca_{21} & ca_{22} & ca_{23}\\
ca_{31} & ca_{32} & ca_{33} \\
\end{bmatrix}
%}
Matrix Multiplication
{%
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33} \\
\end{bmatrix}
\begin{bmatrix}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23}\\
b_{31} & b_{32} & b_{33} \\
\end{bmatrix}
%}
If {% A %} is an {% m \times n %} matrix and {% B %} is an {% n \times s %} matrix, then the
matrix {% C %} which is the result of multiplying {% A \times B %} is a {% m \times s %} matrix
with entries defined by
{% c_{ij} = \sum_{k} a_{ik}b_{kj} %}
A visual representation of the multiplication of A and B is given
{% AB =
\begin{bmatrix}
- a_1 - \\
...\\
- a_n - \\
\end{bmatrix}
\begin{bmatrix}
| & ... & | \\
b_1 & ... & b_m\\
| & ... &| \\
\end{bmatrix}
=
\begin{bmatrix}
a_1^T b_1 & ... & a_1^T b_m \\
& ... &\\
a_n^T b_1 & ... & a_n^T b_m \\
\end{bmatrix}
%}
