Transpose of a Matrix

Transpose of a Matrix

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Vikram
Vikram Singh
Assistant Manager - Content
Updated on Apr 12, 2024 10:56 IST

Transpose of a matrix is a matrix flipped over its main diagonal, switching the matrix’s rows and column indices. In this article, we will briefly discuss how to transform a matrix, its properties and examples.

In math, transposing a matrix is the process of switching its columns and rows. Transposing a matrix changes the location of its elements but doesn’t alter their properties. A transposed matrix is one that has been switched from its original position.

This article will cover everything you need to know about transposing a matrix and how to do it.

Let’s get started!

Table of Content

What Does Transposing a Matrix Mean?

When we talk about transposing a matrix, it means switching the rows and columns of that matrix. Technically, you are not changing the matrix itself but the location of its elements. Transposing a matrix changes its size but not its values.

Definition

Transpose of a matrix is a matrix that is obtained by interchanging the rows and columns.

or
Transpose of a matrix is a matrix flipped over its main diagonal, switching the matrix’s rows and column indices.

Representation

If A is any matrix, then Transpose of A is given by AT or A’.

i.e., if A is any matrix of order m by n, then the order of transpose of A will be n by m.

A = [aij]m x n, then AT = A’ = [aij]n x m .

Example

Here, we have taken two matrices, A and B, of order 3 x 4 and 3 x 3, respectively, and when we transpose them, the order of matrices will be 4 x 3 and 3 x 3, respectively.

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Also Read: All about Symmetric Matrix

How to Transpose a Matrix using Python

In the above examples of transposing a matrix, we have seen that transposing a matrix is nothing but inter-changing the rows and columns (or changing the row to the column and vice versa) of a matrix.
Now, we will see how to transform matrices using Python.

  1. Using NumPy

Code


 
#Python Programm to Transpose a Matrix using Numpy
# import numpy
import numpy as np
#create a matrix using array
matrix = np.array ([[1, 2, 3], [2, 4, 6], [7, 8, 9]])
#using transform function of numpy
matrix.transpose()
Copy code

Output

2. Using Nested Loop


 
#Python Programm to Transpose a Matrix using Nested Loop
#create a matrix
A = [[0, 2, 3],
[-2, 0, 6],
[-3, -6, 0],
[8, 6, 5]]
#create a matrix to store the result
T = [[0,0,0,0], [0,0,0,0], [0,0,0,0]]
# iterate through rows
for x in range(len(A)):
#iterate through columns
for y in range(len(A[0])):
T[y][x] = A[x][y]
for t in T:
print(t)
Copy code

Output

Properties of Transpose of a Matrix

  • Transpose of Transpose of a Matrix

Transpose of Transpose of a matrix is again the same matrix, i.e., for any matrix A.

(AT)T = A

Example

  • Addition

If A and B are two matrices, then the transpose of A + B is equal to the transpose of A + transpose of B, i.e.,

(A + B)T = A T + B T

Example

  • Scalar Multiplication

If a matrix is multiplied by a constant and then transpose is taken, then the result is equal to the transpose of the original matrix multiplied by a constant, i.e.,

(kA) T = kA T

Example

  • Multiplication

If A and B are two matrices, then the transpose of AB is equal to the product of the transpose of A and transpose of B, i.e.,

(AB) T = B T AT

Example:

  • Inverse

If A is an invertible matrix, then the transpose of the inverse of A is equal to the inverse of the transpose of A, i.e.,

(A-1)T = (AT)-1

Example:

  • Determinant

For any square matrix A, the determinant of A is equal to the determinant of the transpose of A, i.e., 

det (A) = det(AT)

Example

Also Read: All About Skew-Symmetric Matrix

 

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Conclusion

In this article, we have briefly discussed the transpose of a matrix with examples in Python. The process of transposing a matrix is simple; the most important thing to remember is that you must interchange rows and columns.

Hope this article will help you to learn the concepts of transposing a matrix.

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FAQs on Transpose of Matrix

What is the Transpose of a Matrix?

Transpose of a matrix is a matrix that is obtained by interchanging the rows and columns.

or
Transpose of a matrix is a matrix flipped over its main diagonal, switching the matrix’s rows and column indices.

How do you find the transpose of a matrix?

To find the transpose of a matrix, rewrite the first row of the matrix as the first column, the second row as the second column, and so on for all rows.

What is the transpose of a rectangular matrix?

The transpose of a rectangular matrix is another rectangular matrix where the number of rows and columns are interchanged. So, if the original matrix has dimensions , the transpose will have dimensions .

How does transposing affect the determinant of a matrix?

The determinant of a matrix remains unchanged upon transposing. That is, .

Can the transpose of a matrix change its rank?

No, the rank of a matrix is invariant under transposition. This means that the rank of is equal to the rank of .

About the Author
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Vikram Singh
Assistant Manager - Content

Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio