Transposition — Definition, Formula & Examples

Transposition is the operation of flipping a matrix over its main diagonal, so that each row becomes a column and each column becomes a row. The result is called the transpose of the original matrix.

Given an $m \times n$ matrix $A$ , its transpose $A^T$ is the $n \times m$ matrix whose entry in row $j$ , column $i$ equals the entry in row $i$ , column $j$ of $A$ . That is, $(A^T)_{ji} = A_{ij}$ for all valid $i, j$ .

Key Formula

(A^T)_{ji} = A_{ij}

Where:

$A$ = The original m × n matrix
$A^T$ = The transpose, an n × m matrix
$i$ = Row index in the original matrix
$j$ = Column index in the original matrix

How It Works

To transpose a matrix, take the first row and write it as the first column, then the second row becomes the second column, and so on. If the original matrix is

m \times n

, the transpose will be

n \times m

. A square matrix that equals its own transpose is called symmetric. Transposition preserves addition and scalar multiplication:

(A + B)^T = A^T + B^T

and

(cA)^T = cA^T

. For products, the order reverses:

(AB)^T = B^T A^T

Worked Example

Problem: Find the transpose of the matrix A = [[1, 2, 3], [4, 5, 6]].

Identify dimensions: A is a 2 × 3 matrix, so its transpose will be 3 × 2.

Swap rows and columns: Write each row of A as a column. Row 1 (1, 2, 3) becomes column 1; row 2 (4, 5, 6) becomes column 2.

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

Answer:

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

Why It Matters

Transposition appears throughout linear algebra and statistics. It is essential for computing dot products as matrix products (

\mathbf{u} \cdot \mathbf{v} = \mathbf{u}^T \mathbf{v}

), forming the normal equations in least-squares regression, and finding orthogonal matrices in physics and computer graphics.

Common Mistakes

Mistake: Reversing only some rows or columns instead of swapping every row-column pair.

Correction: Every entry must move: the element in row i, column j goes to row j, column i. Carefully rewrite each full row of the original as the corresponding column of the transpose.