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

### Eigenvectors of a matrix

#### Example on the eigenvectors 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

Here the given matrix is

Today I’ll do the last part of the problem, that is, to get the eigenvectors of a matrix.

From my earlier post on eigenvalues of any matrix, I already know the eigenvalues of the matrix

These are

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

###### Step 1

For the matrix is

Suppose is the corresponding eigenvector for the eigenvalue

Thus I can say

Now I use matrix multiplications on the left-hand side.

Therefore I get three equations as

Next I will simplify these three equations to get

(1)

(2)

and

(3)

From equations (1) and (2), I get

Now I substitute in equations (3).

Then it becomes

Thus I can write in terms of as

Thus the eigenvector is

Therefore I can say that for the corresponding eigenvector is

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

###### Step 2

For the matrix is

Suppose is the corresponding eigenvector for the eigenvalue

Thus I can say

Now I use matrix multiplications on the left-hand side.

Therefore I get three equations as

Next I will simplify these three equations to get

(4)

(5)

and

(6)

From equation (4), I get

Also from equation (5), I get

Thus the eigenvector is

Therefore I can say that for the corresponding eigenvector is

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

###### Step 3

For the matrix is

Suppose is the corresponding eigenvector for the eigenvalue

Thus I can say

Now I use matrix multiplications on the left-hand side.

Therefore I get three equations as

Next I will simplify these three equations to get

(7)

(8)

and

(9)

From equation (7), I get

That means

Similarly from equation (8), I get

That means

Thus the eigenvector is

Therefore I can say that for the corresponding eigenvector is

Hence I can conclude that for the corresponding eigenvector is ; for the corresponding eigenvector is and for the corresponding eigenvector is

This is the answer to the last part of this example.

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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