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Matrix Diagonalization
• Last Updated : 09 Mar, 2021

Prerequisite:
Eigen values and eigen vectors

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 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 matrx A is related to D, then and the matrix A is reduced to the diagonal matrix D through another matrix P. (P ≡ 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. where,
A -> given 3x3 square matrix.
I -> identity matrix of size 3x3.
λ -> eigen value.
Step 3 - Compute the corresponding eigen vectors using the equation given below. where,
λi -> eigen value.
Xi -> corresponding eigen vector.
Step 4 - Create the modal matrix P. Here, all the eigen vectors till Xi are filled column wise in matrix P.
Step 5 - Find P-1 and then use equation given below to find diagonal matrix D. 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 matrix. [ D = P-1AP ]

Step by step solution:

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.  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 eigen value will replace λ1, λ2 and λ3 in the initial diagonal matrix created in Step 1. Reference articles: Determinant of a matrix & Inverse of a matrix

We know that On solving, we get Putting in Diagonalization of Matrix equation, we get  My Personal Notes arrow_drop_up