In max-plus algebra, matrix multiplication schedules events and the eigenvalue is the cycle time of the steady state. For a two-headed motor, that eigenvalue is the step period — and its eigenvector is the gait.
Max-plus algebra keeps the shape of linear algebra but swaps the two operations. Every "sum" becomes a maximum and every "product" becomes an ordinary sum. That one substitution turns matrix powers into schedules and eigenvalues into periods.
Read an entry A[i][j] as the time it takes for event j to enable event i. A max-plus matrix product then answers a scheduling question rather than an arithmetic one:
Instead of summing products, you take the longest two-step route from j to i. Because a node can't start until all its inputs are ready, the max is the binding constraint.
Raising A to the k-th power in max-plus accumulates delays along paths. The (i, j) entry is the slowest — hence rate-limiting — way to get from j to i in exactly k steps.
A cycle's mean is its total weight divided by its length. The largest such mean is the max-plus eigenvalue: the bottleneck loop that sets the steady-state period. Every other part of the system eventually locks to it.
2-cycle 1→2→1 with weights 2τδ each → mean = (2τδ + 2τδ)/2 = 2τδ. Self-loop τγ → mean = τγ. The winner is max(τγ, 2τδ).
Let Dk(v) be the heaviest length-k walk ending at v (starting anywhere). Sweeping k from 0 to n gives the maximum cycle mean in O(n·edges) — no cycle enumeration. This is exactly what the tool below runs.
The eigenvector matters just as much as the eigenvalue. Its components are the phase offsets of each node within one period. If two events sit half a period apart in the eigenvector, they alternate — and that alternation is a gait.
Edit the matrix, or load a preset. The tool runs Karp's algorithm for the eigenvalue, builds the critical graph, and solves the eigenvector by the Kleene star of the normalized matrix — then reads the eigenvector out as a stepping schedule.
A processive two-headed motor advances only when the trailing head has completed its chemistry and the leading head is bound. Both conditions gating one event is the max operation; delays adding along the mechanochemical path is the plus. The stepping cycle is a max-plus recurrence, so its throughput is a tropical eigenvalue.
Collapsing the reducible transition structure to its recurrent block B isolates the loop that sets the pace. The eigenvalue of B is the step period; whether the τγ self-loop or the 2τδ hand-over-hand cycle dominates is a single comparison, and the crossover is a regime change you can point at on the hero above. The eigenvector then fixes the relative timing of the two heads — the two-beat gait as a consequence of the algebra, not an assumption.
Karp O(n·E)Kleene-star eigenvectorcritical graph All three run client-side in this page — no server, no dependencies — so it drops straight into a talk or onto the site.