Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 2 / 26I / c / Solution

Use the $C$ equation at level $i-1$ to express $h_{C_i}$ in terms of the preceding level, then substitute it into the $W$ equation at level $i$. This yieldsThe standard eigenpair is $\boxed{1,(1,1{)}^T}$. Each row sums to one because constant functions are harmonic for every Markov generator. This recurrence matrix need not have nonnegative entries; it encodes a harmonic equation, rather than a transition matrix. The superscript $(i)$ labels the level vector and is not an additional printed subpart.