Transpose of a Matrix: Definition and Examples

The transpose of a matrix 'A' is denoted as 'A^T' and is obtained by interchanging the rows and columns of 'A'. That is,

$$ \boldsymbol{A}^T=\begin{bmatrix} a_{11} & a_{21} & a_{31} & a_{41}\ a_{12} & a_{22} & a_{32} & a_{42}\ a_{13} & a_{23} & a_{33} & a_{43}\ a_{14} & a_{24} & a_{34} & a_{44} \ \end{bmatrix} $$

For example, if

$$ \boldsymbol{A}=\begin{bmatrix} 1 & 2 & 3\ 4 & 5 & 6 \ \end{bmatrix} $$

then

$$ \boldsymbol{A}^T=\begin{bmatrix} 1 & 4\ 2 & 5\ 3 & 6 \ \end{bmatrix} $$

Note that the dimensions of 'A' and 'A^T' are reversed. That is, if 'A' is an mￗn matrix, then 'A^T' is an nￗm matrix.