Overview
Matrices need not be square. When a matrix has a single row or single column, we refer to them as vectors.
A row vector is simply a single row of matrix
{%
\begin{bmatrix}
a & b & c \\
\end{bmatrix}
%}
A column vector is a matrix with a single column.
{%
\begin{bmatrix}
a \\
c \\
e \\
\end{bmatrix}
%}
Multiplication of a Row Vector and a Column Vector
Multiplication of a row vector by a column vector follows the normal rules of matrix multiplication.
{%
\begin{bmatrix}
a & b & c \\
\end{bmatrix}
\times
\begin{bmatrix}
d \\
e \\
f \\
\end{bmatrix}
= a \times d + b \times e + c \times f
%}
Given a two vectors structured as tuples, {% u = (u_1,u_2,u_3) %} and {% v = (v_1,v_2,v_3) %},
the dot product
{% u \cdot v %}
can be intrepeted as {% u^T v %} (assuming we interpret a tuple vector as a column vector)
Likewise, if a column vector {% v %} is written in Dirac Notation {% | v \rangle %}, and a row vector is a vector in the dual space, written as {% \langle u | %}, then the multiplication can be interpreted as the inner product.
In Physics literature, the components of a column vector are label with an uppercase numeral, {% v^i %}. The elements of a row vector are then specified with a subscript, {% u_i %}. Then the multiplication of a row vector and a column vector is then written as
{% \sum_i u_i v^i %}
or, when using the Einstein convention
{% u_i v^i %}
Multiplication of a Column Vector and a Row Vector
{%
\begin{bmatrix}
d \\
e \\
f \\
\end{bmatrix}
\times
\begin{bmatrix}
a & b & c \\
\end{bmatrix}
=
\begin{bmatrix}
ad & bd & cd \\
ae & be & ce \\
af & bf & cf \\
\end{bmatrix}
%}
This is known as the
outer product, and written in
Dirac Notation
is {% | v \rangle \langle u | %}