QM- matrix exponentiation

1. May 4, 2013

QM--- matrix exponentiation

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

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How do you go about exponentiating a 3*3 matrix? for example if you have

<θ,∅|exp(-i*∅*Ly/h)|l,m>

2. Relevant equations

I know how to exponentiate a two cross two diagonalized matrix. you just exponentiate the diagonal terms. However, in my question, all diagonal entries are zero. So what do I do?

2. May 4, 2013

micromass

Staff Emeritus
Can you give the actual matrix??

I think the best technique is trying to diagonalize the matrix. Then use the result of diagonal matrices.

If the matrix is not diagonalizabe, then you'll need to compute the Jordan canonical form.

3. May 4, 2013

k* 0 1 0
1 0 1
0 1 0

That's the matrix of Ly for l=1.

4. May 4, 2013

micromass

Staff Emeritus
So, what if you try to diagonalize it?

5. May 4, 2013

I will get the eigenvalues 1, -1, 0

6. May 4, 2013

micromass

Staff Emeritus
Can you find a basis of eigenvectors?

7. May 4, 2013

Yes I found the the eigen basis.

8. May 4, 2013

micromass

Staff Emeritus
OK, so you expressed

$A = PDP^{-1}$

with $D$ diagonal.

Now, you need to calculate

$$e^{A} = e^{PDP^{-1}}$$

Now use the definition of the matrix exponential and use that $(PDP^{-1})^k = PD^k P^{-1}$.