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Table of contents
1.
Introduction
2.
Reduce to Diagonal Form
3.
FAQs
4.
Key Takeaways
Last Updated: Mar 27, 2024
Easy

Reduction of Matrix to Diagonal Form

Leveraging ChatGPT - GenAI as a Microsoft Data Expert
Speaker
Prerita Agarwal
Data Specialist @
23 Jul, 2024 @ 01:30 PM

Introduction

Here we will show how to diagonalize a matrix if it is diagonalizable. What do you mean by a matrix?

Matrix is a way of storing the data in rows and columns. Matrices in arrays are used to store data in an organized form.

Let's see further how we can reduce the matrix to its diagonal form.

Reduce to Diagonal Form

We will explain how to diagonalize a matrix if it is diagonalizable.

As an example, we solve the following problem. A concrete example is provided below, and several exercise problems are presented at the end of the article.

The image below depicts the matrix of 2X2, which we have to reduce to diagonal form.

To reduce the matrix ‘A’ in a diagonal form, we have to find the model matrix ‘P’ and inverse of ‘P,’ multiplication of P-1AP will give you the diagonal matrix as shown in the image below.

The next step would be to find out the characteristic polynomial of A. And find the eigenvalues that are 2 and -5, as shown below in the image.

Put these numbers into the diagonal of a matrix with zeroes everywhere else in whichever order you like, and you're done! Yippee!

Let's verify our solution, though.

Firstly, find the eigenvectors associated with the eigenvalues of this matrix.

Let's start with the first eigenvalue λ=2 and plug it into our matrix A−λI, and after that, try to find all of the solutions to  (A−λI)x=0

For case 2 i.e, for λ = -5

Verify your diagonal matrix by constructing a model matrix P of your eigenvectors and using the equation P−1AP=Λ.

Stick your eigenvectors into the columns of a matrix:

Find the inverse P−1 of that matrix, multiply out the expression P−1AP and see if you get back your diagonal matrix.

We got the final matrix as the diagonal matrix

P-1AP = D.

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FAQs

1. What do you mean by a diagonal matrix?

The matrix D is a diagonal matrix (i.e., entries off the main diagonal are all zeros).

2. When did a square matrix say to be diagonalizable?

A square matrix A is diagonalizable if A is similar to a diagonal matrix, i.e., if A = PDP−1 where P is invertible and D is a diagonal matrix.

Key Takeaways

This blog has extensively shown reducing the matrix to diagonal form. A diagonal matrix is an n*n square matrix with non-zero entries only along the diagonal from the under left to the lower right.

More Problems related to the diagonalization of a matrix for the practice are gathered over here.

If you want to practice problems on the matrix you can refer to these links:

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Happy Learning!!!

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