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

If you’re looking for more in eigenvalues and eigenvectors of matrices, do check-in:

**eigenvectors of repeated eigenvalues, **

**complex eigenvalues and eigenvectors of a matrix.**

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**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 the 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. And any matrix is a square matrix if the number of rows and the number of columns of that matrix are the same.

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**

Now here the given coefficient matrix is

And I have to find out its eigenvalues and the eigenvector corresponding to each eigenvalue.

Now 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.

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

**Step 1**

As I have already mentioned above, 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.

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

As I can see, is a common term through out the expression. So I can say that the value of will be

And that gives

Next, I’ll factorise to get

Thus the characteristic equation of the matrix is

Next, I’ll 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.

Now the second part of this example is in my other post namely * eigenvectors of a matrix*.

Dear friends, this is the end of 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!!

Danie Sandova says

Thanks for sharing.

That made me to comprehend the concept of Eigen vectors more easily

Dr. Aspriha Peters says

Thanks a lot. I am even more motivated now to keep going.