Let \(\mathbf{A}\) be an \(m \times n\) matrix. The transpose of \(\mathbf{A}\), denoted by \(\mathbf{A}^{\mathrm{T}}\), is the \(n \times m\) matrix defined by,

\[\left(\mathbf{A}^{\mathrm{T}}\right)_{i,j} = \mathbf{A}_{j,i}\]