partial trace of density matrix

climbon

I am unsure how to (mathematically) do the partial trace of a density matrix so that I can find the expectation value of an observable.

I am working on a model similar to the Jaynes cummings model. My density matrix is of the form;

[tex]
\rho = [\rho_{11}, \rho_{12}, \rho_{21}, \rho_{22}]
[/tex]

As a 2x2 matrix. My system is a composite system as;

[tex]
H_{A} \otimes H_{B}
[/tex]

I want to find the partial trace over the field so I can use an observable M to find the population inversion of the atom;

[tex]
\rho^{A}(t) = Tr_{F}\rho(t)
[/tex]

That way I can;

[tex]
Tr(M \bullet \rho^{A})
[/tex]

To find the inversion of the atom.

How do i do the trace over the field...I understand the principle but struggling how to do this mathematically??

Thanks