Ex. 3 - Define the matrix P˜ whose (i,j) entry is πi 1/2 pijπj -1/2 ,

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Ex. 3 - Define the matrix P˜ whose (i,j) entry is πi 1/2 pijπj -1/2 , i.e., P˜: = D 1/2PD -1/2 , where D ≜ diag{π1, π2, … , πN} is the diagonal matrix with diagonal entry {πi }. Verify that the detailed balance condition is equivalent to the symmetry of P˜ : P˜ ⊤ = P˜.

Ex. 3 - Define the matrix P˜ whose (i,j) entry is πi 1/2 pijπj -1/2 , i.e., P˜: = D 1/2PD -1/2 , where D ≜ diag{π1, π2, … , πN} is the diagonal matrix with diagonal entry {πi }. Verify that the detailed balance condition is equivalent to the symmetry of P˜ : P˜ ⊤ = P˜.

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Ex. 3 - Define the matrix

\tilde{P}

whose

(i, j)

entry is

π_{i}^{1 / 2} p_{i j} π_{j}^{- 1 / 2}

, i.e.,

\tilde{P} := D^{1 / 2} P D^{- 1 / 2},

where

D ≜ diag {π_{1}, π_{2}, \dots, π_{N}}

is the diagonal matrix with diagonal entry

{π_{i}}

. Verify that the detailed balance condition is equivalent to the symmetry of

\tilde{P}

:

{\tilde{P}}^{⊤} = \tilde{P} .

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