

9.3 Spectral decomposition of
the 3D SRHD equations
The full spectral decomposition including the right and left
eigenvectors of the Jacobian matrices associated to the SRHD system
in 3D has been first derived by Donat et al. [75]. Previously, Martí et
al. [179] obtained the spectral
decomposition in 1D SRHD, and Eulderink [83] and Font et
al. [92] derived the eigenvalues
and right eigenvectors in 3D. The Jacobians are given by
where the state vector
and the flux vector
are defined in Equations (6) and (7), respectively. In the
following we explicitly give both the eigenvalues and the right and
left eigenvectors of the Jacobi matrix
only (the cases
are easily obtained by symmetry considerations).
The eigenvalues of matrix
are
and
A complete set of right-eigenvectors is
where
The corresponding complete set of left-eigenvectors is
where
is the determinant of the matrix of
right-eigenvectors, i.e.,
For an ideal gas equation of state
, i.e.,
, and hence
for
.

