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:

whereλeigen values_{1, λ2, λ3 ->}

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,λeigen value._{i ->}Xcorresponding eigen vector._{i ->}

Step 4 -Create the modal matrix P.

Here, all the eigen vectors till X_{i}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^{-1}AP ]

**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 X_{1}, X_{2}, X_{3}corresponding to the eigen values λ = 1,2,3.

Step 5 -Creation of modal matrix P. (here, X_{1}, X_{2}, X_{3}are column vectors)

Step 6 -Finding P^{-1 }and then putting values in diagonalization of a matrix equation.[D = P^{-1AP]}

We do

Step 6to find out which eigen value will replace λ_{1}, λ_{2 }and λ_{3 }in the initial diagonal matrix created inStep 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