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The singular value decomposition (SVD), or equivalently the eigenvalue
decomposition in the case of a real symmetric matrix, for
(1) is relatively easy to perform. We can immediately
write down four eigenvectors:

| |
(26) |

and their corresponding eigenvalues, respectively
2*l*, 2*l*, 2*m*, and *a*-*b* = 2*m*. All four correspond to shear
modes of the system.
The two remaining eigenvectors must be orthogonal to all four of these
and therefore both must have the general form
| |
(27) |

with the corresponding eigenvalue
| |
(28) |

The remaining condition that determines both *X* and is
| |
(29) |

which, after substitution for , leads to a quadratic equation
having the solutions
| |
(30) |

The ranges of values for are
and, since *X*_{-} = -2/*X*_{+},
.The interpretation of the solutions
is simple for the isotropic limit
where *X*_{+} = 1 and *X*_{-} = -2, corresponding respectively to
pure compression and pure shear modes. For all other cases, these two
modes have mixed character, indicating that pure compression cannot be
excited in the system, and must always be coupled to shear. Some
types of pure shear modes can still be excited even in the
nonisotropic cases, because the other four eigenvectors in
(26) are unaffected
by this coupling, and they are all pure shear modes. Pure compressional
and shear modes are obtained as linear combinations of these two mixed
modes according to

| |
(31) |

with for pure shear, and
| |
(32) |

and with for pure compression.
To understand the behavior of *X*_{+} in terms of the layer
property fluctuations or, alternatively, in terms of the Thomsen
parameters, it is first helpful to note that the pertinent functional
is easily shown
to be a monotonic function of its argument *x*. So it is sufficient
to study the behavior of the argument *x* = (*a*+*b*-*c*)/*f*.