Matrix Transpose

Overview

The transpose of a matrix is obtained by interchanging its rows and columns. This operation is fundamental in linear algebra and appears frequently in applications.

Definition

For an $m \times n$ matrix $A$, the transpose $A^T$ is an $n \times m$ matrix where:

$(A^T)_{ij} = a_{ji}$

Row $i$ of $A$ becomes column $i$ of $A^T$.

Notation

$A^T \quad \text{or} \quad A' \quad \text{or} \quad A^t$

Visual Example

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

$A$ is $2 \times 3$, $A^T$ is $3 \times 2$

Properties of Transpose

Property	Formula
Double transpose	$(A^T)^T = A$
Sum	$(A + B)^T = A^T + B^T$
Scalar	$(cA)^T = cA^T$
Product	$(AB)^T = B^T A^T$
Inverse	$(A^{-1})^T = (A^T)^{-1}$

Product Transpose

Important: The transpose of a product reverses the order:

$(ABC)^T = C^T B^T A^T$

Proof for $AB$

$((AB)^T)_{ij} = (AB)_{ji} = \sum_k a_{jk}b_{ki} = \sum_k b_{ki}a_{jk} = \sum_k (B^T)_{ik}(A^T)_{kj} = (B^T A^T)_{ij}$

Transpose and Special Matrices

Symmetric Matrices

$A$ is symmetric if $A^T = A$

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

Skew-Symmetric Matrices

$A$ is skew-symmetric if $A^T = -A$

$\begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0 \end{bmatrix}^T = \begin{bmatrix} 0 & -2 & 3 \\ 2 & 0 & -4 \\ -3 & 4 & 0 \end{bmatrix} = -\begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0 \end{bmatrix}$

Useful Identities

Creating Symmetric Matrices

For any matrix $A$:

• $AA^T$ is symmetric
• $A^T A$ is symmetric
• $A + A^T$ is symmetric
• $A - A^T$ is skew-symmetric

Decomposition

Any square matrix can be written as:

$A = \frac{A + A^T}{2} + \frac{A - A^T}{2}$

where the first term is symmetric and the second is skew-symmetric.

Transpose and Inner Products

For vectors $\mathbf{u}$ and $\mathbf{v}$:

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

For matrices:

$(A\mathbf{x}) \cdot \mathbf{y} = \mathbf{x}^T A^T \mathbf{y} = \mathbf{x} \cdot (A^T \mathbf{y})$

Examples

Example 1: Find Transpose

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

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

Example 2: Verify $(AB)^T = B^T A^T$

$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$

$AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}, \quad (AB)^T = \begin{bmatrix} 19 & 43 \\ 22 & 50 \end{bmatrix}$

$A^T = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}, \quad B^T = \begin{bmatrix} 5 & 7 \\ 6 & 8 \end{bmatrix}$

$B^T A^T = \begin{bmatrix} 5 \times 1 + 7 \times 2 & 5 \times 3 + 7 \times 4 \\ 6 \times 1 + 8 \times 2 & 6 \times 3 + 8 \times 4 \end{bmatrix} = \begin{bmatrix} 19 & 43 \\ 22 & 50 \end{bmatrix} \checkmark$

Example 3: Check Symmetry

$A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}$

$A^T = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}$

Since $A^T = A$, the matrix is symmetric.