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

Here x1 and x2 are column vectors of matrix S (and also eigenvectors of matrix A)

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

Lambda is eigenvalue and x is eigenvector of matrix A

So

This could be expanded further as

L is a diagonal matrix with eigenvalues of matrix A as its non-zero elements.

Matrix L

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.

References

Lec 22 | MIT 18.06 Linear Algebra, Spring 2005

Advanced Engineering Mathematics, 10Th Ed by Erwin Kreyszig

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