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

Would like to read more on eigenvalues and eigenvectors of a matrix? Do check-in:

**Eigenvectors of repeated eigenvalues of a matrix**

**Complex eigenvalues and eigenvectors of a matrix**

**DOWNLOAD READ & PRINT – Eigenvectors of a matrix (pdf)**

**Eigenvectors of a matrix**

**Example of 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**

Now here the given matrix is

And today I’ll do the last part of the problem, that is, to get the eigenvectors of a matrix. Also, from my earlier post on **eigenvalues of any matrix**, I already know the eigenvalues of the matrix are

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

**Step 1**

Now for the matrix is

Also is the identity matrix or unit matrix.

Then I’ll simplify it to get

Now let’s suppose is the corresponding eigenvector for the eigenvalue

Thus I can say

Next, I use matrix multiplications on the left-hand side.

Therefore I get three equations of and . So the first equation is

Then the second equation is

And the last equation is

Now I’ll get the values of and .

**Step 2**

So I will simplify these three equations. As I can see from the first equation, I get

(1)

Now from the second equation also I get the same thing as the first equatiom. And that is

(2)

Finally, I look into the third equation. And that gives me

(3)

As I can see from equations (1) and (2),

Next, I substitute in equation (3). And that gives

Thus I can write in terms of . And that will be

Thus the eigenvector is

Since is a common term, I’ll take it out. Therefore the eigenvector will be

Then I’ll take out as a common term as well. Thus the eigenvector will be

Therefore I can say that for the corresponding eigenvector is

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

**Step 3**

So for , the matrix is

Then I’ll simplify it. And that gives

Again, let’s suppose is the corresponding eigenvector for the eigenvalue

Thus I can say that

Now I use matrix multiplications on the left-hand side. Therefore I get three equations of and . So the first equation is

And the next equation is

Then the last equation is

Now I’ll get the values of and .

**Step 4**

Next, I will simplify these three equations. So the first equation gives

(4)

Then the second equation gives

(5)

(6)

So from equation (4), I get

And that means

Also, from equation (5), I get

So that gives

Thus the eigenvector is

Since is a common term, I can take it out. Therefore the eigenvector will be

Thus I can say that for the corresponding eigenvector is

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

**Step 5**

So for the matrix is

Then I’ll simplify it. And that gives me

So let’s suppose is the corresponding eigenvector for the eigenvalue . Thus I can say that

Now I use matrix multiplications on the left-hand side. Therefore I get three different equations. And the first equation is

Then the next equation will be

And the final equation is

Now I’ll get the values of and .

**Step 6**

So I’ll simplify these three equations. Thus from the first equation, I get

(7)

Then from the second equation, I get

(8)

And finally, from the last equation I get

(9)

So, from equation (7), I get And that means

Similarly, from equation (8), I get Thus I can say that

Thus the eigenvector is

Since is common in all these three rows, I can take it out. So the eigenvector will be

Then I’ll take out as well, since this is also a common term. Thus the eigenvector will be

So I can say that for the corresponding eigenvector is

Hence I can conclude that for the corresponding eigenvector of the matrix is . And for the corresponding eigenvector of the matrix is Finally, for the corresponding eigenvector of the matrix is

And this is the answer to the last part of the given example.

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!!

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