QM - linear algebra

1. Sep 23, 2007

cscott

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

I'm given the Taylor expansion of $e^{i\Theta M}$
Then the question says "Compute the exponential for the given matrix M and verify that the resulting matrix is unitary.

3. The attempt at a solution

I really just don't know what they're asking...

2. Sep 23, 2007

genneth

What's the given matrix? And do you know what unitary means?

3. Sep 23, 2007

cscott

Given matrix [[3/4, 1/4][1/4, 3/4]] and yeah I know what unitary means.

4. Sep 23, 2007

dextercioby

Cscott, is that a hermitian matrix ?

5. Sep 23, 2007

cscott

Yes it is.

I just really don't know what they're asking me to do.

6. Sep 23, 2007

genneth

So if you have the expression for exponentiating a matrix, and the matrix to exponentiate, it seems to me like you can do it. What's the problem?

7. Sep 23, 2007

cscott

They give me the Taylor expansion for that exponential (I was just too lazy to latex it)... wouldn't it be stupid to evaluate a random amount of terms?

$$e^{i\theta M} = \Sigma\frac{\left(i\theta\right)^n}{n!}M^n$$ from n=0 to infinity

8. Sep 23, 2007

cscott

I suppose having the diagonalized version of the matrix would help for raising it to a power.

Is it correct that the diagonalized version would be [[2, 0][0, 1]]?

9. Sep 23, 2007

genneth

If you put M into the form UDU* where U is a unitary matrix, D is diagonal, and U* is the inverse (or transpose) of U, does that help?

And no, that diagonalised form is not quite right.

Last edited: Sep 23, 2007
10. Sep 23, 2007

cscott

Mmm... how about D= [[1, 0][0, 1/2]] ?

Is there a maple command for doing this?

11. Sep 23, 2007

genneth

I don't know about Maple. But it's pretty easy to do by hand -- it's not like a 2x2 matrix is that big...

12. Sep 23, 2007

cscott

Yeah I know but I just wanted a quick way to verify it.

13. Sep 23, 2007

genneth

In the diagonal form, the elements are the eigenvalues. To find the U matrix I mentioned above, you'll need the eigenvectors anyway. So just act M on the eigenvectors and you should end up with the respective eigenvalues. That should be check enough, and quick.

14. Sep 23, 2007

cscott

I'm pretty sure the answer is D= [[1, 0][0, 1/2]] but if I don't know how high n should go how can I get a matrix out of the Taylor expansion they gave me?

15. Sep 23, 2007

genneth

n is for all n: 0 to infinity. D^n for any diagonal matrix should be easy to evaluate. U*U = I for unitary matrices. And remember that matrix multiplication is distributive: AB+AC=A(B+C). Does that help?