I am trying to clarify the following:
Suppose P is the $N\times N$ transition matrix of a finite-dimensional Markov Chain, with invariant distribution given by N-vector $\mu$ (i.e. $\mu^T=\mu^T P$). Define the time-reverse chain with transition matrix $\hat P$ as: $\hat p_{ij}=\frac{\mu_j}{\mu_i} p_{ji}$
Now define the reversible transition matrix $R=(P+\hat P)/2$. It can be shown that this reversible chain has the same invariant measure $\mu$.
Now if we define an inner product space with $<x,y>_\mu=\Sigma x_i y_i \mu_i$, then $R$ is self-adjoint in this space, with real eigenvalues and orthogonal eigenvectors (w.r.t the inner product defined above).
Somehow, numerically, when I try the above computation on a randomly generated Transition matrix, I do not get a self-adjoint R, and the eigenvectors are not orthogonal in the sense described above. Can anyone tell me if there is an error in above statements ?
Edit (in response to Bryon): Let me elaborate on my statements to see if there is any error
Let us define $D_\mu=Diag(\mu)$, a diagonal matrix with $\mu$ on the diagonal. Now define $A=D_\mu^{1/2}RD_\mu^{-1/2}$. It can be checked that A is symmetric matrix.
Then let eigenvectors of A be $\phi_j$, and eigenvectors of R be $f_j$. The relation between the two is: $f_j=D_\mu^{-1/2}\phi_j$
Now consider $\delta_{ij}=\langle \phi_i,\phi_j\rangle=\langle D_\mu^{1/2} f_i,D_\mu^{1/2} f_j\rangle=\langle f_i,f_j\rangle_\mu$
Edit #2 (Response to Did):
The reversibility of R w.r.t $\mu$ implies $R_{ij}\mu_i=R_{ji}\mu_j \:\: \forall i,j$
Now, the self-adjointness condition is $\langle Rx,y\rangle_\mu=\langle x,Ry\rangle_\mu$
L.H.S: $\Sigma_i \Sigma_j R_{ij}x_j y_i \mu_i=\Sigma_i \Sigma_j R_{ji}x_j y_i \mu_j$ (by the reversibility property above).
Then interchanging $i$ and $j$, we get , L.H.S=$\Sigma_i \Sigma_j R_{ij}x_i y_j \mu_i$, which is the same as R.H.S.
So the definition of inner-product I wrote seems to work here. I am still confused.
Final Edit: I feel stupid writing this, but i have tracked down my error to a faulty piece of Matlab code: apparently the command Eigs (based on Arnoldi algorithm to find top k eigenvalues) gives wrong results (such as complex eigenvalues/vectors). I found the whole spectrum using the other command "Eig" and it turns out R has real spectrum, and orthogonal eigenvectors w.r.t to the inner-product I described.
You should divide, not multiply, by $\mu_i$ in the inner product: use $\langle x,y\rangle_\mu=\sum x_i y_i /\mu_i$.