Eigenvalues of a matrix. Hello friends, today it’s all about the eigenvalues of a matrix. Have a look!!

### Eigenvalues of a matrix

Suppose I have a matrix, say . Also, it’s a matrix.

That means it has 3 rows and 3 columns.

Now if is an eigenvalue of the matrix , then it will satisfy an equation

This equation is called characteristic equation of the matrix .

Here is a matrix. Therefore will have 3 values.

If it would have been matrix or matrix, then would have 4 or 5 values respectively.

Note: It is only possible to get the eigenvalues of any matrix if it’s a square matrix.

Now I will give an example.

#### Example of the eigenvalues of a matrix

Disclaimer: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the example.

##### Example

According to Stroud and Booth (2013)* “For the coefficient matrix …, determine the eigenvalues and an eigenvector corresponding to each eigenvalue:

”

##### Solution

Today I’ll only do the first part of the problem, that is, to get the eigenvalues of a matrix.

I have already shown the second part, that is, the corresponding eigenvectors on another post.

Here the given matrix is

Now, to get the eigenvalues of any matrix, the first task is to get its characteristic equation. So, I’ll start with that.

###### Step 1

I have already mentioned above that the characteristic equation of any matrix is

Here is the eigenvalue of the matrix . is the unit matrix or the identity matrix.

Now means the determinant of the matrix

Thus the matrix is

Therefore the characteristic equation of the matrix is

Now I will simplify the left-hand side of the equation.

That means I’ll expand the determinant.

Therefore it will be

Next, I’ll evaluate it to find

So I’ll simplify it a bit more to get

Thus the characteristic equation of the matrix is

Now the next step is to get the eigenvalues of this matrix.

###### Step 2

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

Now in this example, the characteristic equation is

Thus the solutions are

Hence I can conclude that the eigenvalues of the given matrix are and respectively.

This is the first part of the example.

You will find the second post on my other post on eigenvectors of a matrix.

Dear friends, this is the end of my today’s post. 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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