Diagonalize any matrix. Today I’ll show how to diagonalize any matrix. Have a look!!

Want to check out more related posts on eigenvalue problems? Here you go!!

**Eigenvectors corresponding to each of the eigenvalues of a matrix**

**Diagonalize any matrix**

Now a diagonal matrix means a matrix with diagonal elements only. And All other elements are zero in that matrix. For example

is a diagonal matrix but

is not a diagonal matrix.

Now suppose I have a matrix, like

And I have to diagonalise it.

Note: the diagonal matrix of any matrix is the matrix with its eigenvalues at the diagonal positions.

For example, if the matrix has four eigenvalues as , then its diagonal matrix will be

But if I have to diagonalize a matrix, say , I should follow certain steps. And these are:

- First of all, I’ll get its eigenvalues and their corresponding eigenvectors.
- Next, I’ll write the matrix with the eigenbases and get the matrix .
- Then I’ll use the formula for the diagonal matrix as .

Now I’ll solve an example for you.

**Solved example of how to diagonalize any matrix**

Disclaimer: This is not my own example. I have chosen it from a book. I have also given the due reference at the end of the post.

So here is the example.

**Example 1**

According to Kreyszig (2005)*, “Find the eigenbases and then diagonalise the following matrix:

”

**Solution**

Now here the given matrix is

And I have to find out its eigenbases and then to diagonalise it. First of all, I’ll find out the eigenvalues of the matrix .

**Step 1**

Now the characteristic equation of the matrix is

Then I will simplify the left-hand side of the equation. And that means I have to **expand the determinant**.

Therefore it will be

Next, I’ll evaluate it. So what I get is

Now I’ll simplify it. And that gives

Thus the characteristic equation of the matrix is

As I know it already, the eigenvalues of any matrix are the solutions of the characteristic equation of the matrix. Therefore the eigenvalues of the matrix are

Now I’ll get the eigenvector corresponding to each eigenvalue.

**Step 2**

First of all, I’ll get the eigenvector corresponding to the eigenvalue .

Thus for the matrix is

Suppose is the corresponding eigenvector for the eigenvalue

So I can say

Now I use matrix multiplication to get a system of equations as

Thus I can say that

Also from , I can say that

Therefore I can say that

where is any number.

Thus the eigenvector is

Therefore I can say for , the corresponding eigenvector is

Now I’ll get the eigenvector corresponding to the eigenvalue .

**Step 3**

Thus for the matrix is

Suppose is the corresponding eigenvector for the eigenvalue

Then I’ll use matrix multiplication to get a system of equations as

Thus I can say that

Also from , I can say that

Therefore I can say that

where is any number.

Thus the eigenvector is

Therefore I can say for , the corresponding eigenvector is

Now I’ll get the eigenvector corresponding to the eigenvalue .

**Step 4**

Thus for the matrix is

Suppose is the corresponding eigenvector for the eigenvalue

Then I’ll use matrix multiplication to get a system of equations as

Thus I can say that

If I choose , then that values of and will be

Thus the eigenvector is

Therefore I can say for , the corresponding eigenvector is

Now I’ll get the eigenbases for the matrix . So it will be

Next, I’ll get the inverse of the matrix .

**Step 5**

And for that, I’ll use the **Gauss-Jordan method to find out the inverse of a matrix**.

As per the Gauss-Jordan method, I’ll include the unit matrix on the right-hand side like

And my aim is to bring the unit matrix on the left-hand side. As per the Gauss-Jordan method, the matrix on the right-hand side will be the inverse of the matrix .

First of all, I’ll subtract row from row . Also, I’ll add twice row to row .

So in mathematical form, it will be:

Then the resultant matrix is

As I can see, the unit matrix is on the left-hand side of the matrix .

So this means the right-hand side matrix is the inverse of the matrix .

Thus the inverse of the matrix is

As I know, the diagonal matrix is

Next, I’ll get the value of .

**Step 6**

Now I’ll use the matrix multiplication to get the value of . And that means

Then I’ll get the value of . Thus it will be

Thus the diagonal matrix is

Hence I can conclude that I have got the answers for this example.

Dear friends, this is the end of today’s post on how to diagonalize any matrix. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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