# Finding eigenvalues, Shankar exercise 1.8.3

1. Oct 25, 2010

### TimID

First, I appologise if this is in the wrong place, while the book is QM, the question is pure maths. Also I'm not sure if this techically counts as homework as I am self studying. Finally, sorry for the poor formatting, I'm not that good with LaTeX

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

Given the matrix: $$\Omega$$ =
$$\left[ {\begin{array}{ccc} 2 & 0 & 0 \\ 0 & 3 & -1 \\ 0 & -1 & 3 \\ \end{array} } \right]$$

Show that $$\omega$$1 = $$\omega$$2 = 1; $$\omega$$3 = 2

2. Relevant equations

3. The attempt at a solution

So det($$\Omega$$ - $$\omega$$I) = (2 - $$\omega$$)((3 - $$\omega$$)(3 - $$\omega$$) - 1)

Which obviously leaves $$\omega$$ = 2, but also (3 - $$\omega$$)2 = 1, the solutions to which should be $$\omega$$ = 2 and $$\omega$$ = 4.

Where am I going wrong?

Any help greatfully appreciated.

2. Oct 25, 2010

### vela

Staff Emeritus
You're not going wrong for the given matrix. Are you sure you're doing the right problem?

3. Oct 26, 2010

### TimID

You're absolutely right, there was a factor of a half in the original question that I completely missed, thanks. I must have checked the original problem a dozen times before posting and didn't spot it, I hate my brain sometimes.

Thanks again,

Tim