A particular decomposition is the Jordan
canonical form which is for a matrix
**X**

where
**J _{i}**

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One needs to calculate the exponential of the transition rate matrix
(see(3.19)). Due to the special structure of
with main diagonal elements
it can be shown by Gershgorin's
circle theorem that all its eigenvalues lie in the left half-plane and one
eigenvalue is zero (see left side of Fig. 3.8).
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It is known that Krylov subspace methods tend to provide good approximations for the eigenvalues located in the outermost part of the spectrum [98]. Since all eigenvalues of lie in the left half-plane, the eigenvalues which dominate the exponential function, , are located in the inner part of the spectrum. Thus the eigenvalue spectrum of has to be transformed to make the eigenvalues which dominate the exponential function coincide with the eigenvalues which are filtered out by the Krylov subspace method. An inversion is not possible since is singular. One can move the eigenvalue spectrum to the right with (see right side of Fig. 3.8). In addition the upper limit of the spectral radius derived from Gershgorin's disc theorem is reduced by a factor of two. Since the commutative law applies, one may write

The transformed matrix
has its eigenvalues which
dominate the exponential function in the outermost part of its spectrum.
Thus, a Krylov subspace method will give good results even for small
dimension subspaces.

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The objective is the computation of an approximation of the form
to the matrix exponential operation
,
where
is a
polynomial
of degree m-1 in
,
which is a linear combination of the
vectors
,
and thus is an element of the Krylov subspace [98]
Constructing an orthonormal basis
in the Krylov subspace,
and choosing
,
one may write using the
identity
B*

where

which still involves the evaluation of the exponential of a matrix, but this time of small dimension

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