## partial trace of density matrix

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;

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

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

$$H_{A} \otimes H_{B}$$

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;

$$\rho^{A}(t) = Tr_{F}\rho(t)$$

That way I can;

$$Tr(M \bullet \rho^{A})$$

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
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