How to diagonalize a matrix? Example of diagonalization

Diagonalization of a matrix

How to diagonalize a matrix? Example of diagonalization. An nxn matrix A is said to be diagonalizable if there exists an invertible matrix P such that A=PDP.

Procedure for diagonalizing a matrix

For diagonalizing a matrix A, the first step is to find the eigen values of it. Then find the corresponding eigen vectors of it. All the eigen vectors should be linearly independent if you want to diagonalize a matrix A. Otherwise, A is not diagonalizable. If you get the distinct eigen values then by the theorem eigen vectors corresponding to distinct eigen values are linearly independent. But if the eigen values are repeated, then the corresponding eigen vectors can be or cant be linearly independent. The diagonal matrix we obtain after the process contains the eigen values in the main diagonal.

Example of diagonalizing a matrix.

The first example shows how we got repeated eigen value, but since the corresponding eigen vectors are linearly independent so A is diagonalizable. But in the second case again with the repeated eigen values, there were only existing two eigen vectors so that is why those vectors were linearly dependent. Hence that matrix was not diagonalizable.

example of diagonalization of matrix 1
example of diagonalization of matrix 1

 

diagonalization of matrix 1 (2)
diagonalization of matrix 1 (2)
diagonalization of matrix 1 (3)
diagonalization of matrix 1 (3)

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Example of diagonalization

Example of diagonalization of a matrix 

 

Also read here

https://eevibes.com/mathematics/linear-algebra/what-are-the-similar-matrices/

How to perform similar matrices transformation?

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