Let = (ij ) be an m m matrix and = (i) be a 1 m row vector. Show that the

Let Λ = (Λij ) be an m × m matrix and π = (πi) be a 1 × m row vector. Show that the equality πiΛij = πjΛji is true for all pairs

(i, j) if and only if diag(π)Λ = Λt diag(π), where diag(π) is a diagonal matrix with ith diagonal entry πi. Now suppose Λ is an infinitesimal generator with equilibrium distribution π. If P(t) = etΛ

is its finite-time transition matrix, then show that detailed balance

πiΛij = πjΛji for all pairs (i, j) is equivalent to finite-time detailed balance πipij (t) = πjpji(t) for all pairs (i, j) and times t ≥ 0.