# Homework Help: Calculating the eigenvalues of an interesting matrix

1. Nov 13, 2012

### JohnLang

1. The problem statement, all variables and given/known data

Let T be a square nxn matrix and let each entry tij=n(i-1)+j.

Calculate the eigenvalues of T of T when T is a 3x3 matrix and a 4x4 matrix.

2. Relevant equations

3. The attempt at a solution

For the homework I'm supposed to calculate the 3x3 case and the 4x4 case. Then it says find the eigenvalues for the general nxn case for extra credit.From matlab I'm able to see that the matrix T has rank 2 (If this helps). I'm sort of stuck on this part.

On MatLab when I've done up to order 6, each from order 3,4,5,6 have 2 non-zero eigenvalues and the other eigenvalues are 0 with algebraic multiplicity n-2

Last edited: Nov 13, 2012
2. Nov 13, 2012

### JohnLang

I feel like I need to exploit some properties of determinants.

3. Nov 13, 2012

### I like Serena

Welcome to PF, JohnLang!

To find the eigenvalues typically you need to solve the characteristic polynomial.
Have you tried to find the characteristic polynomial?

You can already observe some of its properties.
Since the rank of each matrix is 2, what does that mean for the form of the characteristic polynomial?

In particular you will need a way to calculate the second and third coefficient of the characteristic polynomial.
Do you know how to find those?

Btw, can you deduce that the rank is always 2?
To find the rank of a matrix, the typical way to do it, is to find how many independent column vectors there are.
This can be done by repeatedly subtracting one column from the other until no more columns become zero.

4. Nov 13, 2012

### JohnLang

Hi, I Like Serena!

Thanks for welcoming me!

Anyway, so far I think I was able to show that the rank of the matrix which is 2 I used a really rough method

so I think what this means for the characteristic polynomial, it's going to look something like this

p(t)=tn-2(t2-bt+c)

I'm kinda stuck on finding the coefficients I'm not quite sure. I calculated the tr(A)=(n3+n)/2

5. Nov 13, 2012

### Robert1986

I'd try to see if you can somehow figure out a general way to row-reduce it into its Jordan Form or into a triangular form.

6. Nov 13, 2012

### JohnLang

Robert:

So when I row reduced this matrix it looks something like this

[1 0 -1 -2 -3 ... 1-n;0 1 2 3 4 5 ... n-1;(remaining rows are 0)

which means that this matrix has rank 2

from here what should I do?

Also since the matrix T has rank 2, this means that there are non zero principal (2x2) minors from the referred literature from I Love Serena. I think that's how it's worded.

7. Nov 13, 2012

### I like Serena

Yep. That looks correct.
In particular you have b=tr(A) and c is the sum of the principle minors.
And I see you already found a general formula for the trace of A.
Good!

That leaves a general formula for the sum of the principle minors.
But for the first part you don't need a general formula yet, you don't even have to use principle minors.
You just need the value of c for the 3x3 and the 4x4 matrix.
With that you can solve the characteristic polynomial and find the eigenvalues.

That's enough.
This proves that the rank is 2, meaning the characteristic polynomial has the form you already mentioned.

What's left is to find b and c in all cases and solve it for the eigenvalues.

After that you can try to find a general formula for c.
But let's take one step at a time.

8. Nov 13, 2012

### I like Serena

Well, your problem asks for the eigenvalues of the 3x3 matrix and the 4x4 matrix that are filled with the consecutive numbers 1, 2, 3, etcetera.

So I mean these 2 cases.

Btw, to find the pattern, I'd also do the 2x2 matrix.

9. Nov 13, 2012

### JohnLang

I'm having problems when I'm calculating 4x4 matricies, my principal 3x3 minors are all 0
which makes sense since the rank of A is 2 so in this case rref(A) will have 2 rows of zeros which implies the determinant of A is 0.

Any further suggestions?

10. Nov 13, 2012

### I like Serena

Ah, but the principal minors are not zero.

Can you calculate the following determinant and reorder it into a polynomial?
Then you should see what "c" is in this case.
$$\text{det} \begin{bmatrix} 1 - λ & 2 & 3 \\ 4 & 5 - λ & 6 \\ 7 & 8 & 9 - λ \end{bmatrix}$$
If you can, perhaps you can try to keep track how you effectively calculate c?

11. Nov 13, 2012

### Robert1986

If this is the correct row reduction (I suspect there is an error somewhere because the first row has n+1 entries) then it *might* be easy to find two eigenvectors that have different eigenvalues. These will be the two eigenvalues you need (note the eigenvectors are not the eigenvectors of the original matrix.) And, as you said, there are only two non-zero eigenvalues.

12. Nov 13, 2012

### JohnLang

so when I do it this way I get the characteristic polynomial, p(λ)=-λ(λ2-15λ-18)

and that gives me the correct eigenvalues

the principal minors are

[5 6;8 9]

[1 3;7 9]

[1 2;4 5]

the sum of the principal minors give me -18, furthermore I see that the trace is b. but when I do the 4x4 case, the principal minors of the 4x4 matrix are

[6 7 8;10 11 12; 14 15 16]

[1 3 4;9 11 12;13 15 16]

[1 2 4;5 6 8;13 14 16]

[1 2 3;5 6 7;9 10 11]

and all their determinants are 0

I'm saying that the 4x4 char poly is

p(λ)=λ22-65λ-C)

I guess I'm lost on the C part.

13. Nov 13, 2012

### I like Serena

The 3x3 case is correct.

But the trace of the 4x4 is not 65.
And the principal minors are supposed to be 2x2 matrices (much easier to calculate ;)).

14. Nov 13, 2012

### JohnLang

sorry 34, also if i'm gonna calculate the principal minors for 2x2 that means there will be 6 of them for the 4x4 case right? 4 choose 2?

15. Nov 13, 2012

### I like Serena

Yup.

16. Nov 13, 2012

### JohnLang

WOOT! ok the 4x4 case worked, let's see if I can get it going for the general case, at least I know there are n choose 2 principal 2x2 minors

17. Nov 13, 2012

### I like Serena

18. Nov 13, 2012

### JohnLang

is it fair to say that only the first 2 rows of T will be considered in the general case of finding the 2x2 principal submatricies since the rank of T is 2 meaning there are a lot of zero rows?

19. Nov 13, 2012

### I like Serena

Hmm, I don't know.
I don't think so.

Anyway, I'm off to bed. :zzz:

20. Nov 13, 2012

### JohnLang

haha thanks for all the help!

21. Nov 14, 2012

### Dick

No, but you can take an alternate route skipping the whole submatrix thing that's kind of like that. Since you know that the rank is 2, if you presume there are two nonzero eigenvalues then their eigenvectors span the range R of T. Further T(R)=R. So you can figure out everything you want to know by analyzing how T maps the two dimensional space R to R. How easy that is depends on how clever you can be about picking a basis for R.

22. Nov 14, 2012

### JohnLang

Hmm, this is interesting, since T has 2 distinct eigenvalues in which I'll presume, furthermore since the other eigenvalues are all zero, is this enough to say that T is diagonalizable? Would that help in me in analyzing how T maps R?

23. Nov 14, 2012

### Dick

That's too abstract to really help. You do have to get your hands dirty. Take the 3x3 case. You want to pick two basis vectors for R that are easy to work with. Try messing around.

24. Nov 14, 2012

### JohnLang

btw each of the non zero eigenvalues have a corresponding eigenvector because geometric multiplicity is always less than or equal to algebraic multiplicity right?

25. Nov 14, 2012

### Dick

Well, every eigenvalue has at least one eigenvector. But I wouldn't worry too much about proving these things exist. The idea here is to actually find the eigenvalues and potentially eigenvectors. At which point there really won't be any question they exist.