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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}


\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}