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### Multiplicative Reversible Version of the Transition Matrix; Spectral Representation

At first we will discuss a method enabling us to transform (ergodic) transition matrices such that the resulting matrix is reversible.

• Let be an irreducible and aperiodic (but not necessarily reversible) transition matrix and let be the corresponding stationary initial distribution such that for all .
• Moreover, we consider the stochastic matrix where (99)

i.e., where is also an irreducible and aperiodic transition matrix having the same stationary initial distribution .
• The pair , where the stochastic matrix is given by , is reversible as we observe Definition The matrix is called the multiplicative reversible version of the transition matrix .

Remarks

• All eigenvalues of are real and in because has the same eigenvalues as the symmetric and nonnegative definite matrix , where and hence • As a consequence, the symmetric matrix is diagonalizable and the right and left eigenvectors and can be chosen such that
• for all • the vectors are an orthonormal basis in .
• Then and , where and (100)

are right and left eigenvectors of , respectively, as for every  and This yields the following spectral representation of the multiplicative reversible version obtained from the transition matrix ; see also the spectral representation given by formula (30).

Theorem 2.15 For arbitrary and  (101)

where and are the right and left eigenvectors of defined in .

Proof

• As the (right) eigenvectors of defined in (100) are also a basis in , for every there is a (uniquely determined) vector such that • Furthermore, we have and hence for arbitrary and .
• Thus we obtain • On the other hand, (100) implies for arbitrary and  (102)

where the last equality takes into account that for all and that the eigenvectors von are an orthonormal basis of .
• This proves the spectral representation (101).     Next: Alternative Estimate for the Up: Reversibility; Estimates for the Previous: Determining the Rate of   Contents
Ursa Pantle 2006-07-20