📘

Matrix Transpose

Implement a function to compute the transpose of a given matrix, a fundamental operation in linear algebra. This task involves swapping the rows and columns of the input matrix.

The concept of a matrix transpose is crucial in various mathematical and computational contexts, as it allows for the transformation of a matrix $A$ into its transpose $A^T$ , effectively flipping the matrix over its diagonal. If $A$ is an $m \times n$ matrix, then $A^T$ is an $n \times m$ matrix, where the element at position $(i, j)$ in $A^T$ is equal to the element at position $(j, i)$ in $A$ .

Understand the dimensions of the input matrix $A$ .
Create a new matrix $A^T$ with swapped dimensions.
Populate $A^T$ with elements from $A$ , using the relationship $(A^T)_{ij} = A_{ji}$ .

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

This technique is widely used in computer vision and machine learning for data transformation and matrix operations.

Example:

Input:

A = [[1, 2, 3],
     [4, 5, 6]]

Output:

[[1, 4], [2, 5], [3, 6]]

Reasoning:

Step-by-step transformation using the transpose definition:

$A^T_{ij} = A_{ji}$

Original matrix $A$ (2×3): $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$

Columns become rows:
- Column 0 $[1, 4]$ → Row 0
- Column 1 $[2, 5]$ → Row 1
- Column 2 $[3, 6]$ → Row 2
Equivalently, flip across the diagonal: $A_{ij} \rightarrow A^T_{ji}$
- $A_{00}=1 \rightarrow A^T_{00}=1$
- $A_{01}=2 \rightarrow A^T_{10}=2$
- $A_{02}=3 \rightarrow A^T_{20}=3$
- $A_{10}=4 \rightarrow A^T_{01}=4$
- $A_{11}=5 \rightarrow A^T_{11}=5$
- $A_{12}=6 \rightarrow A^T_{21}=6$
Result $A^T$ (3×2): $A^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$

Constraints:

Matrix dimensions: 1 ≤ m, n ≤ 100
Elements are numbers

Editor

Python 3.13.1

Test Results

0/0

Run code to see test results.