# Diagonal bases in transformations

1. Sep 7, 2011

### derryck1234

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

Let T: R3 - R3 be the linear operator given by

T = -y + z
-x + z
x + y

Find a basis B' for R3 relative to which the matrix for T is diagonal using the standard basis B for R3.

2. Relevant equations

[T]B' = P-1[T]BP

3. The attempt at a solution

I find the standard matrix for T to be

0 -1 1
-1 0 1
1 1 0

The characteristic equation of which, I find to be

(lambda)^3 -3(lambda) + 2 = 0

Which has no real solutions? What can I do?

Thanks

Derryck

2. Sep 7, 2011

### lanedance

well the matrix is symmetric so that should ensure real eigenvalues...

from visual inspection it appears 1 is a root

3. Sep 7, 2011

### tiny-tim

Hi Derryck!

(have a lambda: λ and try using the X2 icon just above the Reply box )
erm

how can a cubic equation have no real solutions?

4. Sep 7, 2011

### lanedance

I also got the same characteristic equation as well...

5. Sep 7, 2011

### derryck1234

Ok thanks guys. See the thing is I just put it into excel to help me find roots. I must have entered the wrong formula though:( It came up with an irrational number? Anyway...I can definitely see that 1 is a root now...thanks...