The Transpose of a Matrix

Definition: If $A$ is an $m \times n$ matrix, then the Transpose of $A$ denoted $A^T$ is the $n \times m$ matrix resulting from interchanging both the rows and columns of $A$, that is $(A)_{ij} = (A^T)_{ji}$.

For example, suppose that we have the following $3 \times 4$ matrix $A = \begin{bmatrix}3 & 0 & 2 & 1\\ 2 & 3 & 1 & 7\\ -2 & 3 & 1 & 4 \end{bmatrix}$. If we interchange the rows and columns of $A$, then we will obtain the $4 \times 3$ matrix $A^T = \begin{bmatrix} 3 & 2 & -2\\ 0 & 3 & 3\\ 2 & 1 & 1\\ 1 & 7 & 4 \end{bmatrix}$.

One important point to note is that if $A$ is a square matrix, then the diagonal entries of $A^T$ will be the same as that of $A$. For example, if $A = \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}$, then $A^T = \begin{bmatrix} 1 & 4 & 7\\ 2 & 5 & 8\\ 3 & 6 & 9 \end{bmatrix}$. Notice that the entries in the main diagonal are the same between both matrices.

Example 1

Given the following matrix, determine $G^T$.

(1)

\begin{align} G = \begin{bmatrix} 2 & 1 & 3 & 2 & 1\\ 4 & 9 & 1 & 0 & 3 \end{bmatrix} \end{align}

There are 2 rows and 5 columns to G, hence G is a $2 \times 5$ matrix, and $G^T$ will be a $5 \times 2$ matrix. Interchanging the rows and columns of G we obtain:

(2)

\begin{align} G^T = \begin{bmatrix} 2 & 4\\ 1 & 9\\ 3 & 1\\ 2 & 0\\ 1 & 3 \end{bmatrix} \end{align}

Example 2

Prove that $G + G^T$ is only defined if $G$ is a square matrix.

• Proof: Suppose not, and let G be a matrix of size $m \times n$ where $m \neq n$. It thus follows that $G^T$ will be of size $n \times m$. Recall that matrix addition is only defined if all matrices in the sum are the same size. Hence $G_{m \times n} + G^T_{n \times m}$ is not defined since $m ≠ n$. $\blacksquare$