Transpose of a Matrix

Tags: #◇CSMATH2 #◇Linear Algebra #◇Computational Mathematics #◇Matrices #◇Matrix Operations #◇Computer Science #◇Vector Spaces #◇Linear Transformations

---

◇Overview

Definition: The transpose of an $m \times n$ matrix $A=[a_{ij}]$, denoted $A^T$, is the $n \times m$ matrix obtained by interchanging the rows and ...

◇Key Concepts

Definition Example

◇Details / Theory

Definition: The transpose of an $m \times n$ matrix $A=[a_{ij}]$, denoted $A^T$, is the $n \times m$ matrix obtained by interchanging the rows and columns of $A$. If $A^T=[a'_{ij}]$, then $a'_{ij}=a_{ji}$. (Handout 1, Page 27)
PN: Think of it as "flipping" the matrix across its main diagonal (from top-left to bottom-right). The first row becomes the first column, the second row becomes the second column, and so on. Example: If $A=\begin{bmatrix} 1 & 2 & 3 \\ 0 & -1 & 4 \end{bmatrix} (size 2 \times 3), then A^T=\begin{bmatrix} 1 & 0 \\ 2 & -1 \\ 3 & 4 \end{bmatrix}$ (size $3 \times 2$). (Handout 1, Page 27)

◇Common Pitfalls

$(AB)^T = B^T A^T$ — the order reverses! $(A^T)^T = A$ — double transpose returns the original. The transpose of an $m \times n$ matrix is $n \times m$.

◇Summary

TL;DR — $A^T$ swaps rows and columns: $(A^T)_{ij} = a_{ji}$. Key property: $(AB)^T = B^T A^T$. A matrix equals its transpose iff it is symmetric.

◇Links

Related: Source / Reference:

◇References & Navigation

• Previous Topic: ◇Trace of a Matrix
• Next Topic: ◇Triangular Matrices
• Related Notes: ◇Important Remarks, ◇Powers of a Matrix, ◇Matrices Detailed