QM Linear Algebra: Compute Matrix Exponential & Verify Unitarity

[cscott]

Homework Statement

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.

The Attempt at a Solution

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

 

[genneth]

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

 

[cscott]

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

 

[dextercioby]

Cscott, is that a hermitian matrix ?

 

[cscott]

Yes it is.

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

 

[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?

 

[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

 

[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]]?

 

[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.

 

[cscott]

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

Is there a maple command for doing this?

 

[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...

 

[cscott]

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...

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

 

[genneth]

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

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.

 

[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?

 

[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?