Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE) PDF Download

Introduction

Let A and B be two matrices of order n. B can be considered similar to A if there exists an invertible matrix P such that B=P^{-1} A P  This is known as Matrix Similarity Transformation.

Diagonalization of a matrix is defined as the process of reducing any matrix A into its diagonal form D. As per the similarity transformation, if the matrix A is related to D, then

D = P^-1 AP  and the matrix A is reduced to the diagonal matrix D through another matrix P. Where P is a modal matrix) 

Modal matrix: It is a (n x n) matrix that consists of eigen-vectors. It is generally used in the process of diagonalization and similarity transformation.

In simpler words, it is the process of taking a square matrix and converting it into a special type of matrix called a diagonal matrix.

Steps Involved:

Step 1: Initialize the diagonal matrix D as:

where λ₁, λ₂, λ₃ -> eigen values

Step 2: Find the eigen values using the equation given below.
det(A-λI)=0        
where, A -> given 3×3 square matrix. I -> identity matrix of size 3×3. λ -> eigen value.

Step 3: Compute the corresponding eigen vectors using the equation given below.

 
where, λ_i -> eigen value. X_i -> corresponding eigen vector.

Step 4: Create the modal matrix P.

Here, all the eigenvectors till Xi have filled column-wise in matrix P. 

Step 5: Find P^-1 and then use the equation given below to find diagonal matrix D.
D=P^-1 AP 

Example Problem

Problem Statement: Assume a 3×3 square matrix A having the following values:

Find the diagonal matrix D of A using the diagonalization of the matrix. [ D = P^-1AP ]
Step 1: Initializing D as:

Step 2: Find the eigen values. (or possible values of λ)


Step 3: Find the eigen vectors X₁, X₂, X₃ corresponding to the eigen values λ = 1,2,3.

 

On solving, we get the following equation
x₃ = 0 (x₁)x₁ + x₂ = 0


Similarly, for λ = 2

and for

similarly

Step 5: Creation of modal matrix P. (here, X₁, X₂, X₃ are column vectors)

Step 6: Finding P^-1 and then putting values in diagonalization of a matrix equation. [D = P^-1AP]
We do Step 6 to find out which eigenvalue will replace λ₁, λ₂, and λ₃ in the initial diagonal matrix created in Step 1.



Since det(P) ≠ 0 ⇒ Matrix P  is invertible
we know that

On solving, we get

Putting in the Diagonalization of Matrix equation, we get