Matrix Transpose

What is a Matrix Transpose?

The transpose of a matrix A, denoted as A^T, is the matrix formed by turning the rows of A into columns (or equivalently, turning the columns of A into rows). For an m × n matrix A, its transpose A^T is an n × m matrix.

The Matrix Transpose Formula

For a matrix A, its transpose A^T is defined as:

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

Where:

• (A^T)_{ij} is the element at the i-th row and j-th column of A^T
• A_{ji} is the element at the j-th row and i-th column of A

Calculation Steps

1. Identify the dimensions of the original matrix A (m × n)
2. Create a new matrix A^T with dimensions n × m
3. For each element in A^T, copy the corresponding element from A, swapping the row and column indices

Example Calculation

Let's calculate the transpose of matrix A:

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

1. Identify the dimensions: A is a 2 × 3 matrix

2. Create A^T with dimensions 3 × 2

3. Copy elements, swapping row and column indices:

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

Visual Representation

This diagram visually represents how the elements of matrix A are rearranged to form its transpose A^T. The arrows show how each element moves from its position in A to its new position in A^T.