The **matrix transpose** $A^T$ of $A$ is the matrix obtained by every element $a_{ij}$ with $a_{ji}$ $A^T=(a_{ji}) \qquad\iff\qquad A=(a_{ij})$ If $A$ is $m\times n$ then $A^T$ is $n\times m$, since the entries that used to be aligned in a [[row vector]] in $A$ are aligned in a [[column vector]] in $A^T$ A matrix equal to its transpose is a [[symmetric matrix]]