Matrix Transpose: Flipping Across the Diagonal

Definition

The transpose of an $m \times n$ matrix $A$, written $A^T$, is the $n \times m$ matrix obtained by reflecting $A$ across its main diagonal. Equivalently: $(A^T)_{ij} = A_{ji}$

Row $i$ of $A$ becomes column $i$ of $A^T$, and vice versa.

Step-by-step

$A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ (2 rows × 3 columns).

• Row 1 of $A$: $(1, 2, 3)$ → column 1 of $A^T$.
• Row 2 of $A$: $(4, 5, 6)$ → column 2 of $A^T$.

$A^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$

Shape flips: $A$ is $2 \times 3$, $A^T$ is $3 \times 2$. Main diagonal entries (positions where $i = j$) stay the same; off-diagonal entries swap.

Key properties

• Double transpose: $(A^T)^T = A$.
• Transpose of a sum: $(A + B)^T = A^T + B^T$.
• Transpose of a product (swap order): $(AB)^T = B^T A^T$.
• Transpose of inverse: $(A^{-1})^T = (A^T)^{-1}$.
• Transpose of a scalar multiple: $(cA)^T = c A^T$.

The swap in $(AB)^T = B^T A^T$ surprises people. A quick sanity check: if $A$ is $2 \times 3$ and $B$ is $3 \times 4$, then $AB$ is $2 \times 4$ so $(AB)^T$ is $4 \times 2$. On the other side: $A^T$ is $3 \times 2$ and $B^T$ is $4 \times 3$, so $B^T A^T$ is $4 \times 2$. Dimensions match — and the identity holds entry-by-entry.

Special kinds of matrices

• Symmetric: $A^T = A$. Only square matrices can be symmetric. Example: $\begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$.
• Skew-symmetric: $A^T = -A$. All diagonal entries are zero.
• Orthogonal: $Q^T Q = I$, so $Q^T = Q^{-1}$.

Where transpose appears

• Dot products: $\mathbf{u} \cdot \mathbf{v} = \mathbf{u}^T \mathbf{v}$ when vectors are columns.
• Normal equations in least squares: $A^T A \mathbf{x} = A^T \mathbf{b}$.
• Gram matrix: $A^T A$ — symmetric, positive semidefinite, central to SVD and QR.
• Quadratic forms: $\mathbf{x}^T A \mathbf{x}$ appears in optimization and statistics.
• Data science: "rows as samples / columns as features" vs. "rows as features / columns as samples" differ by transposition.

Common mistakes

• Forgetting the shape flip. A $2 \times 3$ matrix transposes to a $3 \times 2$, not another $2 \times 3$.
• Mis-ordering the product transpose. $(AB)^T$ is $B^T A^T$, not $A^T B^T$.
• Assuming diagonal entries change. On a square matrix, diagonal positions $(i, i)$ are fixed — the swap is only off-diagonal.

Try it in the visualization

A 2×3 matrix is drawn as a grid. Each cell flips across the main diagonal, swapping positions $(i, j) \leftrightarrow (j, i)$. Elements animate to their new positions while shape changes from wide to tall.