How to exponentiate a 3*3 matrix with all diagonal entries equal to zero?

[rsaad]

QM--- matrix exponentiation

Homework Statement

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

<θ,∅|exp(-i*∅*Ly/h)|l,m>

Homework 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?

 

[micromass]

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.

 

[rsaad]

k* 0 1 0
1 0 1
0 1 0

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

 

[micromass]

So, what if you try to diagonalize it?

 

[rsaad]

I will get the eigenvalues 1, -1, 0

 

[micromass]

Can you find a basis of eigenvectors?

 

[rsaad]

Yes I found the the eigen basis.

 

[micromass]

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 $(PDP^{-1})^k = PD^k P^{-1}$.