Transpose of a matrix

Intuition

The transpose of a matrix is a new matrix obtained by interchanging the rows and columns of the initial matrix.

For example,
$$\begin{pmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \end{pmatrix}^{T} = \begin{pmatrix} 1 & 4\\ 2 & 5\\ 3 & 6 \end{pmatrix}.$$

Formal definition

Let
$$A=(a_{ij})_{1\le i\le m,\ 1\le j\le n}$$be an $m\times n$ matrix. The transpose of $A$, denoted by $A^T$, $A^t$, or ${}^tA$, is the $n\times m$ matrix
$$A^T=(a_{ji})_{1\le i\le n,\ 1\le j\le m}.$$