# 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

Ais diagonalized to MatrixL

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.