Practice question · Multiple choice
A market-share simulation is run from three wildly different starting splits and all three converge to the same final split. What guarantees that?
Hints
- Decompose the starting vector into eigenvector components. What happens to each after many steps?
- Ask what the steady state depends on: where you began, or how customers switch?
Show the answer
B. The dominant eigenvector, which suppresses every other component
Why
Every component is scaled by its eigenvalue each step, so the largest survives and the rest die away, the destination belongs to the switching behaviour, not the history. It is why churn matrices predict long-run share and why PageRank works on exactly the same mathematics.
Practise Eigenvalues and Eigenvectors
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