Finding a Non-Diagonalizable 3x3 Matrix with 2 as its Only Eigenvalue

[tengxiaona]

Homework Statement

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

Homework Equations

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 I am stuck here

can anyone help me to slove these equtions.

i don't konw the way i doing this problem is right or wrong.

if u have any others ideas,please help me

 

[matt grime]

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?

 

[tengxiaona]

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?

sorry, i don't have any ideas

could u just give me any hints?

 

[Dick]

Hint: think about upper triangular matrices.

 

[tengxiaona]

Hint: think about upper triangular matrices.

but the matrix A is not diagonalizable

 

[Dick]

but the matrix A is not diagonalizable

Do you really think all upper triangular matrices are diagonalizable?

 

[tengxiaona]

Do you really think all upper triangular matrices are diagonalizable?

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
fd+af+ad=12
adf=18

then....=.=

i still can't slove these equtions

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

 

[Dick]

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.

 

[matt grime]

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.

 

[tengxiaona]

i got it now !

i really went to a wrong way!

thanks for DICK and MG you guys help!