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The Hessians ,
The Hessian of
is defined as the matrix or operator of second derivatives
|
(154) |
For functional (109)
and fixed we find the Hessian by taking the derivative
of the gradient in (127) with respect to again.
This gives
|
(155) |
or
|
(156) |
The addition of the diagonal matrix
=
can result in a negative definite
even if has zero modes,
like for a differential operator
with periodic boundary conditions.
Note, however, that
is diagonal and therefore symmetric,
but not necessarily positive definite, because can be negative
for some . Depending on the sign of
the normalization condition for that can be replaced
by the inequality or .
Including the -dependence of and with
|
(157) |
i.e.,
|
(158) |
or
|
(159) |
we find, written in terms of ,
The last term, diagonal in , has dyadic structure in ,
and therefore for fixed at most one non-zero eigenvalue.
In matrix notation the Hessian becomes
the second line written in terms of the probability matrix.
The expression is symmetric under
,
,
as it must be for a Hessian and as can be verified using the symmetry of
and the fact that
and commute, i.e.,
.
Because functional is invariant under a shift transformation,
, the Hessian has
a space of zero modes with the dimension of .
Indeed,
any -independent function
(which can have finite
-norm only in finite -spaces)
is a left eigenvector of
with eigenvalue zero.
Thus, where necessary,
the pseudo inverse of have to be used instead of the inverse.
Alternatively, additional constraints on can be added
which remove zero modes,
like for example boundary conditions.
Next: Gaussian prior factor for
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Joerg_Lemm
2001-01-21