# Find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigenvalu

1. Apr 5, 2009

### tengxiaona

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

Find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigenvalue of A

2. Relevant equations

3. The attempt at a solution

since λ=2,and it is a 3*3 matrix

i get the det(λI-A)=(λ-2)^3=0

then λ^3-6λ^2+12λ-8=0

now i use http://image.cramster.com/answer-board/image/cramster-equation-2009452223286337456700835550002161.gifas[/URL] matirx A

and i calculate

det(λI-A)
=λ^3-λ^2(i+e+a)+λ(ai+ae+ie-cg-hf-bd)-(aef+bfg+cdh+cge-ahf-bdi)=0

now we can easy to know that

i+e+a=6

ai+ae+ie-cg-hf-bd=12

aef+bfg+cdh+cge-ahf-bdi=18

now im stuck here

can anyone help me to slove these equtions.

i dont konw the way i doing this problem is right or wrong.

if u have any others ideas,plz help me

Last edited by a moderator: Apr 24, 2017
2. Apr 6, 2009

### matt grime

Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

There are infinitely many solutions to those equations: 9 variables and 3 equations. Also, solutions to them tell you nothing about whether or not the matrix can be diagonalised. You're going to have to think of a different method.

Do you know of any other forms a matrix can be put into if it is not diagonlisable?

Do you know of any matrices where it is easy to see what the eigenvalues are?

3. Apr 6, 2009

### tengxiaona

Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

sorry, i dont have any ideas

could u just give me any hints?

4. Apr 6, 2009

### Dick

Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

Hint: think about upper triangular matrices.

5. Apr 6, 2009

### tengxiaona

Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

but the matrix A is not diagonalizable

6. Apr 6, 2009

### Dick

Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

Do you really think all upper triangular matrices are diagonalizable?

7. Apr 6, 2009

### tengxiaona

Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

ok,i trid again and stuck again

i set A as

a b c
0 d e
0 0 f

use the way i did before above

finally i get

a+d+f=6

then.................=.=

i still cant slove these equtions

Dick , wut exactly do u mean by ues an upper matrix

8. Apr 6, 2009

### Dick

Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

By an upper triangular matrix I mean exactly what you posted. You should be able to tell me what the eigenvalues of that matrix are without working too hard. That tells you what values a, d and f should have. Now you just want to find an EXAMPLE of values of b, c and e that is NOT diagonalizable. Hint: it's not diagonalizable if you can't find three linearly independent eigenvectors. Just try guessing. Hint: b=c=e=0 is a bad guess.

9. Apr 7, 2009

### matt grime

Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

Where did those 3 equations come from, and why? I think you need to work out the determinant of an upper triangular matrix again. You still haven't addressed what it means for a matrix to be (or not be) diagonalizable, either.

10. Apr 8, 2009

### tengxiaona

Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

i got it now !!!

i really went to a wrong way!!

thanks for DICK and MG you guys help!!!