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.
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.
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.
We know that
On solving, we get
Putting in Diagonalization of Matrix equation, we get