Transpose¶

Definition¶

The transpose of a matrix $\boldsymbol{A}=(a_{ij})$ of size $m\times n$ is the $n\times m$ matrix $\boldsymbol{A}^{\top}$ whose entries are $(\boldsymbol{A}^{\top})_{ij}=a_{ji}$$a_{ji}$.

Interpretation¶

Transposition flips a matrix across its main diagonal: row $i$ of $\boldsymbol{A}$ becomes column $i$ of $\boldsymbol{A}^{\top}$, so an $m\times n$ matrix becomes an $n\times m$ matrix.

The rows $(1,2,3)$ and $(4,5,6)$ of the $2\times3$ matrix become the two columns of the $3\times2$ transpose. The entry $6$ in position $(2,3)$ moves to position $(3,2)$, which is the rule $(\boldsymbol{A}^{\top})_{ij}=a_{ji}$$a_{ji}$, while the diagonal entries $1$ and $5$ stay in place. Transposing twice gives the original matrix back.

The reversed order in $(\boldsymbol{A}\boldsymbol{B})^{\top}=\boldsymbol{B}^{\top}\boldsymbol{A}^{\top}$ is forced by the sizes. If $\boldsymbol{A}$ is $m\times n$ and $\boldsymbol{B}$ is $n\times k$$k$, then $\boldsymbol{B}^{\top}\boldsymbol{A}^{\top}$ is a $(k\times n)(n\times m)$ product and is defined, while $\boldsymbol{A}^{\top}\boldsymbol{B}^{\top}$ is a $(n\times m)(k\times n)$ product, which is not defined unless $m=k$$k$. Writing $(\boldsymbol{A}\boldsymbol{B})^{\top}=\boldsymbol{A}^{\top}\boldsymbol{B}^{\top}$ is therefore a common slip.

Transposition turns a column vector into a row vector, which is how row vectors are usually written. A matrix is symmetric when $\boldsymbol{A}=\boldsymbol{A}^{\top}$${\mathbf{A}}^{\top }$. Later, the product $\boldsymbol{x}^{\top}\boldsymbol{y}$ of a row and a column becomes the dot product. Other texts write the transpose with a superscript capital T or with a prime; this wiki uses the $\top$ symbol throughout.

Key Results¶

• $(\boldsymbol{A}^{\top})^{\top}=\boldsymbol{A}$$\mathbf{A}$.
• $(\boldsymbol{A}\boldsymbol{B})^{\top}=\boldsymbol{B}^{\top}\boldsymbol{A}^{\top}$ whenever the product is defined.

Examples¶

• If $\boldsymbol{A}=\begin{pmatrix}1&2\\3&4\end{pmatrix}$$\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$, then $\boldsymbol{A}^{\top}=\begin{pmatrix}1&3\\2&4\end{pmatrix}$$\left(\begin{array}{cc}1& 3\\ 2& 4\end{array}\right)$.
• The transpose of $\begin{pmatrix}1&2&3\end{pmatrix}$ is $\begin{pmatrix}1\\2\\3\end{pmatrix}$.

Prerequisites¶

• Matrix

Related Concepts¶

• Inner Product
• Matrix Product
• Vector

Sources¶

• Math II: Linear Algebra · Week 3

Aliases: matrix transpose, transposed matrix

Last updated: 20 October 2026 03:02