2.5. Transpose

Definition - Transpose of a Matrix

The transpose of a matrix is a new matrix that is obtained by swapping the rows and columns of the original matrix. The transpose of a matrix can be thought of as flipping the matrix over its main diagonal, which is the line that goes from the top left corner to the bottom right corner of the matrix. We denote the transpose of a matrix \(A\) by \(A^T\).

The transpose of a matrix is a new matrix that is obtained by swapping the rows and columns of the original matrix. The transpose of a matrix can be thought of as flipping the matrix over its main diagonal, which is the line that goes from the top left corner to the bottom right corner of the matrix. We denote the transpose of a matrix \(A\) by \(A^T\). For example, if

\[\begin{split} A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \end{split}\]

then

\[\begin{split} A^T = \begin{bmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{bmatrix} \end{split}\]

Lemma: Properties of the Transpose of a Matrix

Some of the properties of the tTranspose of a matrix [Kuttler and Farah, 2020, Nicholson, 2018]:

• The transpose of the transpose of a matrix is the original matrix. That is, $\((A^T)^T = A.\)$

• The transpose of the product of two matrices is the product of the transposes of the matrices in reverse order. That is,\((AB)^T = B^TA^T\).

• The transpose of the linear combination of two matrices is the linear combination of the transposes of the matrices with the same coefficients. That is, \((rA + sB)^T = rA^T + sB^T.\)

Example: The transpose of matrix \(A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}\) can be found as follows

\[\begin{equation*} A^T=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}^T=\begin{bmatrix} 1 & 4\\ 2 & 5\\ 3 & 6 \end{bmatrix}. \end{equation*}\]

Example: Find the \(A\) if

\[\begin{align*} \left(A+3 \begin{bmatrix} 1 & -1 & 0 \\ 1 & 2 & 4 \end{bmatrix}\right)^T=\begin{bmatrix} 2 & 1 \\ 0 & 5 \\ 3 & 8 \end{bmatrix}. \end{align*}\]

Solution: We have,

\[\begin{align*} &\left(A+\begin{bmatrix} 3 & -3 & 0\\ 3 & 6 & 12 \end{bmatrix}\right)^T=\begin{bmatrix} 2 & 1 \\ 0 & 5 \\ 3 & 8 \end{bmatrix}. \\ {\text{Transpose both sides}}\quad \Rightarrow \quad &\left(\left(A+\begin{bmatrix} 3 & -3 & 0\\ 3 & 6 & 12 \end{bmatrix}\right)^T\right)^T=\begin{bmatrix} 2 & 1 \\ 0 & 5 \\ 3 & 8 \end{bmatrix}^T. \\ \Rightarrow \quad & A+\begin{bmatrix} 3 & -3 & 0\\ 3 & 6 & 12 \end{bmatrix}=\begin{bmatrix} 2 & 0 & 3\\ 1 & 5 & 8 \end{bmatrix}. \\ \Rightarrow \quad & A=\begin{bmatrix} 2 & 0 & 3\\ 1 & 5 & 8 \end{bmatrix}-\begin{bmatrix} 3 & -3 & 0\\ 3 & 6 & 12 \end{bmatrix}= \begin{bmatrix}-1 & 3 & 3\\ -2 & -1 & -4\end{bmatrix}. \end{align*}\]