Part 24 : Diagonalization and Similarity of Matrices
Diagonalization is a process of converting a n x n square matrix into a diagonal matrix having eigenvalues of first matrix as its non-zero elements.
If we say that
Matrix A is diagonalized to Matrix L
what it really means is that for a matrix A of order n x n there exists another matrix L such that
One more thing to note is that matrix L is also of order n x n.
Example of Diagonalization
Suppose that matrix A is a square matrix of order 2 x 2 with 2 distinct eigenvalues (lambda1 & lambda2) and eigenvectors (x1 and x2).
On arranging all the linearly independent eigenvectors of matrix A as columns in another matrix S, we get
Matrix S should be non-singular as we have to find its inverse later.
On multiplying matrix A with matrix S
We can infer from this
We know that
So
This could be expanded further as
L is a diagonal matrix with eigenvalues of matrix A as its non-zero elements.
Multiplying inverse of matrix S on both the sides of equation
We can also rearrange the equation as
Similarity of Matrices
If this equation holds then matrix A and matrix L will termed as similar.
Eigenvalues of both the matrices will be equal. Also, if x is an eigenvector of matrix A then
will be the eigenvector of matrix L.
Property of Diagonalization
Lets find the square of matrix A
This could be generalized for any power m given m > 0.
