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

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
- 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

are right and left eigenvectors of , respectively, as for every

- All eigenvalues
of
are real and in because
has the same
eigenvalues as the symmetric and nonnegative definite matrix
, where

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).

**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

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).

- As the (right) eigenvectors
of
defined in (100) are also a basis in
, for every
there is a (uniquely
determined) vector
such that