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:
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)

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.

Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE) 
where, λi -> eigen value. Xi -> corresponding eigen vector.

Step 4: Create the modal matrix P.
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
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:

Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
Find the diagonal matrix D of A using the diagonalization of the matrix. [ D = P-1AP ]

Step 1: Initializing D as:
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
Step 2: Find the eigen values. (or possible values of λ)
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
Step 3: Find the eigen vectors X1, X2, X3 corresponding to the eigen values λ = 1,2,3.
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE) 
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
On solving, we get the following equation
x3 = 0 (x1)x1 + x2 = 0
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
Similarly, for λ = 2
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
and for
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
similarly
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
Step 5: Creation of modal matrix P. (here, X1, X2, X3 are column vectors)
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
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.
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
Since det(P) ≠ 0 ⇒ Matrix P  is invertible
we know that
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
On solving, we get
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)
Putting in the Diagonalization of Matrix equation, we get
Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE)

The document Matrix Diagonalization | Engineering Mathematics - Civil Engineering (CE) is a part of the Civil Engineering (CE) Course Engineering Mathematics.
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FAQs on Matrix Diagonalization - Engineering Mathematics - Civil Engineering (CE)

1. What is matrix diagonalization?
Ans.Matrix diagonalization refers to the process of finding a diagonal matrix that is similar to a given matrix. It involves finding a matrix P such that P^(-1)AP is a diagonal matrix, where A is the given matrix.
2. How is matrix diagonalization useful?
Ans.Matrix diagonalization is useful in various areas of mathematics and science, particularly in linear algebra. It simplifies calculations involving matrix powers, matrix exponentials, and matrix equations. It also helps in understanding the properties and behavior of the given matrix.
3. Can every matrix be diagonalized?
Ans.No, not every matrix can be diagonalized. A matrix can only be diagonalized if it satisfies certain conditions. One of the conditions is that the matrix should have n linearly independent eigenvectors, where n is the size of the matrix. If the matrix does not meet these conditions, it cannot be diagonalized.
4. How do you diagonalize a matrix?
Ans.To diagonalize a matrix, follow these steps: 1. Find the eigenvalues of the matrix by solving the characteristic equation. 2. For each eigenvalue, find its corresponding eigenvectors. 3. If the matrix has n linearly independent eigenvectors, form a matrix P by placing the eigenvectors as columns. 4. Calculate P^(-1) and form the diagonal matrix D by placing the eigenvalues along the main diagonal. 5. Finally, the diagonalized matrix is given by P^(-1)AP = D.
5. Can a matrix have multiple diagonalizations?
Ans.Yes, a matrix can have multiple diagonalizations. If a matrix has repeated eigenvalues, it can have different sets of linearly independent eigenvectors associated with each eigenvalue. Therefore, different choices of eigenvectors can lead to different diagonalizations of the same matrix.
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