Section 3.4 Transpose
The transpose of a matrix is obtained by interchanging rows and columns. In terms of components,
\begin{equation}
\left(A_{ij}\right)^T=A_{ji}\text{.}\tag{3.4.1}
\end{equation}
For example,
\begin{equation}
A = \left(\begin{array}{cc}
a\amp b\\
c\amp d
\end{array} \right)
\Longrightarrow
A^T= \left(\begin{array}{cc}
a\amp c\\
b\amp d
\end{array} \right)\tag{3.4.2}
\end{equation}
and
\begin{equation}
B = \left(\begin{array}{ccc}
a\amp b\amp c\\
d\amp e\amp f\\
g\amp h\amp i
\end{array} \right)
\Longrightarrow
B^T= \left(\begin{array}{ccc}
a\amp d\amp g\\
b\amp e\amp h\\
c\amp f\amp i
\end{array} \right)\text{.}\tag{3.4.3}
\end{equation}
A square matrix is called symmetric if it is equal to its transpose, that is, if \(A = A^T\text{.}\)
Non-square matrices also have transposes, for example
\begin{equation}
v = \left(\begin{array}{c}
x\\
y\\
z
\end{array} \right)
\Longrightarrow
v^T= \left(\begin{array}{ccc}
x\amp y\amp z
\end{array} \right)\text{.}\tag{3.4.4}
\end{equation}
