Home Statistical Dictionary Matrix Transpose

Matrix Transpose

The matrix formed by flipping a matrix over its diagonal, swapping rows and columns -- the operation behind $\mathbf{X}'\mathbf{X}$, quadratic forms, and the definition of a symmetric matrix.

Linear Algebra for StatisticsBeginnerFree

In Plain English

Transposing a matrix just means turning its rows into columns and its columns into rows, like rotating a spreadsheet 90 degrees. It sounds trivial, but the transpose shows up constantly in statistical formulas because pairing a matrix with its own transpose, as in $X^{'} X$ , is exactly how you compute sums of squares and cross-products across every pair of variables in one compact expression.

Definition

The transpose of an $m \times n$ matrix $A$ , denoted $A^{⊤}$ or $A^{'}$ , is the $n \times m$ matrix obtained by reflecting $A$ across its main diagonal, so that $(A^{⊤})_{ij} = a_{ji}$ , row $i$ of $A$ becomes column $i$ of $A^{⊤}$ . The transpose satisfies $(A^{⊤})^{⊤} = A$ , $(A + B)^{⊤} = A^{⊤} + B^{⊤}$ , and, importantly, reverses order under multiplication, $(AB)^{⊤} = B^{⊤} A^{⊤}$ ; a matrix that equals its own transpose, $A = A^{⊤}$ , is called symmetric, the defining property of every covariance and correlation matrix.

Formula

(A^{⊤})_{ij} = a_{ji}

Notation

A^{⊤} or A^{'}

—the transpose of A

a_{ji}

—the (j,i) entry of A, placed at position (i,j) in A-transpose

Worked Example

For $A = (142536)$ (a 2x3 matrix), the transpose is the 3x2 matrix $A^{⊤} = 123456$ : the first row of A, (1,2,3), becomes the first column of $A^{⊤}$ .

Properties

Transposing reverses the order of a matrix product, $(AB)^{⊤} = B^{⊤} A^{⊤}$ , not $A^{⊤} B^{⊤}$ -- a rule that must be applied correctly whenever a matrix formula is transposed, algebra errors from getting this order wrong are among the most common mistakes when deriving matrix-based statistical results by hand.
A square matrix equal to its own transpose is symmetric -- every covariance matrix, correlation matrix, and matrix of the form $X^{'} X$ is automatically symmetric, since $(X^{'} X)^{⊤} = X^{'} (X^{'})^{⊤} = X^{'} X$ , which is why so many matrices encountered in statistics have this property built in rather than as a coincidence.
The transpose of a column vector is a row vector and vice versa, so a dot product between two vectors $x$ and $y$ is often written as the matrix product $x^{⊤} y$ , unifying vector and matrix notation under one consistent multiplication rule.

At a Glance

NotationA^T or A' -- rows and columns swap

Dimension changean m x n matrix becomes n x m

Reverses order in products(AB)^T = B^T A^T

A = A^T definesa symmetric matrix

Last updated September 8, 2026← Back to the Dictionary