The transpose of a matrix exchanges rows and columns. If $A$ is an $m\times n$ matrix, then $A^T$ is $n\times m$ and

$$(A^T){ij}=A{ji}.$$

For example,

$$A=\begin{pmatrix}1&2&3\4&5&6\end{pmatrix} \quad\Rightarrow\quad A^T=\begin{pmatrix}1&4\2&5\3&6\end{pmatrix}.$$

Transposing twice returns the original matrix:

$$(A^T)^T=A.$$

A square matrix is symmetric when

$$A^T=A.$$

For a symmetric matrix, entries mirror across the main diagonal: $A_{ij}=A_{ji}$.

Symmetric matrices arise naturally in Hessians, covariance matrices and quadratic forms, where the relationship between two coordinate directions is unchanged when their order is swapped.