Up: The art of fitting

# Appendix A

The purpose of this appendix is to show that using a proper matrix , the noise covariance matrix of Eq. (37) given by:
 (A1)
can have a diagonal form. The matrix can be diagonalized and we can write.
 (A2)
where is diagonal and is an orthogonal matrix (). Replacing Eq. (A2) into Eq. (A1), we have:
 (A3)
Since is positive definite all its eigenvalues are positive, therefore the square root of is defined. Therefore if we apply the following transformation to the data:
 (A4)
we can rewrite Eq. (A3) as:
 (A5)
where is the identity matrix. So replacing in Eq. (33) by will have the effect of removing the artificial correlation due to the noise, and also of performing a normalization. We should point out that the transformation matrix that can achieve this is not unique, and any multiplication by an orthogonal matrix will achieve this. Nevertheless, we give a solution to the problem which can be solved as an eigenvalue and eigenvector problem. The transformation given above does not remove the artificial correlation due the p modes but more or less preserve it. This can have some useful application when one wants to produce spectra with uncorrelated noise but with correlated p-mode signals.

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