Since the matrices and (), where is the diagonal matrix of the eigenvalues of , are trivially calculated once the eigenvalues of are known, we may take advantage of the diagonal form of those matrices to considerably simplify the calculation of for the different forms of the source function considered in the previous section. In fact, if is the matrix whose columns are the eigenvectors of , so that , the following equations hold:

We then consider the equations, related to Eqs. (5), (10), and
(13) respectively,

which determine three diagonal matrices. It is then clear that
we can recover the emerging Stokes vector from the above
equations, observing that

This last equation shows that only the first column of the matrix
is actually needed.

The procedure for calculating has then essentially been reduced to the following steps:

- Determine the matrix (or ) at each point of a suitable grid which subdivides the frequency (or wavelength) domain of the line which is investigated.
- Determine the eigenvalues and the eigenvectors of the matrix (or ), the first yielding the diagonal matrix (and any related function ), and the second giving the transformation matrix .
- Determine (the first column of) .

In previous papers (CL93, Casini & Landi Degl'Inno- centi
1996, hereafter referred to as CL96) we dealt with the determination
of the propagation matrix in the presence of simultaneous electric
and magnetic fields. Following the notation established in those papers,
the propagation matrix can be written, in the absence of continuum
contribution, as (cf. CL96, Eq. (23))

The entries of the matrix are then defined by the set of equations (cf. CL96, Eqs. (16))

, enumerating the four Stokes parameters .

The quantities are the relative strengths of the polarization components (at the frequencies ) of the line. These strengths, and the frequencies , are dependent on the intensities and on the relative geometry of the external fields, and they must be calculated according to the general procedure described in CL93. The quantities and are, respectively, the real and the imaginary part of the complex line profile which is assumed to hold for the line under investigation.

Having considered only thermal-Doppler broadening of the line, the
complex line profile is shown to be

where *w*(*z*) is the function of complex variable (see Abramowitz &
Stegun 1965)

and the parameter *K* is the inverse of the Doppler broadening (in
angular frequency units). For real values of its argument, *w*(*z*) then
becomes

where
*D*(*x*) is the Dawson integral, to be computed numerically.

Once the matrix has been determined, through the methods
described in CL93 (or CL96, in the weak-field case), one needs to
determine the corresponding diagonal matrix and the
transformation matrix . Due to the particular form of the
propagation matrix, it is possible to provide the general algebraic
expressions of its eigenvalues and eigenvectors. For instance, it is
not difficult to demonstrate that the four eigenvalues of the matrix
are given by the formula (see Landi Degl'Innocenti & Landi
Degl'Innocenti 1985)

where we introduced the formal vectors
and
. In Eq. (23) all the possible combinations of the
signs are understood, giving the four eigenvalues.

Quite more complicated formulas for the coefficients of the matrices and can also be found in the literature (e.g., Sidlichovský 1976).

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