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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˜.

related to stochastic process dtmc model

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Ex. 3 - Define the matrix P ~ whose ( i , j ) entry is π i 1 / 2 p i j π j − 1 / 2 , i.e.,
P ~ := D 1 / 2 P D − 1 / 2 ,
where D ≜ diag ⁡ { π 1 , π 2 , … , π N } is the diagonal matrix with diagonal entry { π i } . Verify that the detailed balance condition is equivalent to the symmetry of P ~ :
P ~ ⊤ = P ~ .

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