The transpose of an \(m \times n\) matrix \(A\), denoted by \(A^\mathsf{T}\), is the \(n \times m\) matrix such that the \((j,i)\)-entry is given by \(A_{i,j}\) for \(i = 1,\ldots,m\) and \(j= 1,\ldots, n\). In other words, column \(i\) of \(A^\mathsf{T}\) comes from row \(i\) of \(A\).
Let \(A=\begin{bmatrix} 1 & 2 & 3\\4 & 5 &6\end{bmatrix}\). Then \(A^\mathsf{T} = \begin{bmatrix} 1 & 4 \\ 2 & 5\\ 3 & 6\end{bmatrix}\).
Let \(A = \begin{bmatrix} 7 \\ 8 \\ 9 \end{bmatrix}\). Then \(A^\mathsf{T}=\begin{bmatrix} 7 & 8 & 9\end{bmatrix}\).
The following properties hold:
\((A^\mathsf{T})^\mathsf{T} = A\)
If \(\lambda\) is a scalar, then \((\lambda A)^\mathsf{T} = \lambda A^\mathsf{T}\)
\((A^{-1})^\mathsf{T} = (A^\mathsf{T})^{-1}\)
\((A+B)^\mathsf{T} = A^\mathsf{T}+B^\mathsf{T}\) for any \(B\) having the same dimension as \(A\)
If \(A\) is \(m\times p\) and \(B\) is \(p \times n\), then \((AB)^\mathsf{T} = B^\mathsf{T}A^\mathsf{T}\).
\(\begin{bmatrix} 7 & 2 \end{bmatrix} -\begin{bmatrix} 3 \\ 4 \end{bmatrix}^\mathsf{T}\)
\(\left(\begin{bmatrix} 1 & 2 \\ 5 & 6 \end{bmatrix}^\mathsf{T} + 2\begin{bmatrix} -1 & 1 \\ 1 & 0 \end{bmatrix}\right)^\mathsf{T}\)