Markov Chain Mobile Market Share: Eigenvalues, Diagonalization, and Long-Run Distribution

Published on October 28, 2026

Linear-Algebra

A two-company mobile market share problem modeled by a transition matrix, covering solving linear systems, finding eigenvalues and eigenvectors, verifying Cayley-Hamilton to compute the inverse, diagonalizing to get A^n, and finding the long-run steady state. The approach uses matrix powers and eigenvector decomposition to predict shares over time.

Question

Mobile Network Market Share In a city, people can use only Company A or Company B for mobile service. Every month: 80% of A's customers stay and 20% switch to B. 30% of B's customers switch to A and 70% stay. Let xₙ = (Aₙ, Bₙ)ᵀ be the market shares in month n. Then xₙ₊₁ = A xₙ, where A = [[0.8, 0.3], [0.2, 0.7]] Questions (a) Next month's shares are (0.55, 0.45)ᵀ. Find this month's shares by solving a system of linear equations. (b) Find the eigenvalues and eigenvectors of A. (c) Verify Cayley–Hamilton and use it to find A⁻¹. (d) Diagonalize A and find Aⁿ. (e) If the current shares are (0.5, 0.5)ᵀ, find the shares after n months and the long-run market share.

Answer

Solution

(a) Write

with

and solve

for

, which gives this month’s shares:

Substituting

into the first equation:

, so

and

. The second equation checks:

This month’s shares are

(b) Characteristic polynomial:

and

For : , so (up to scaling).
For : , so .

and

, so the Cayley–Hamilton polynomial is

Since

, we get

(d) Let

and

. Then

and

. Then

with

Check at

: this gives

✓.

(e) Expand

in the eigenvectors:

Therefore

Check:

gives

✓. Since

, the shares converge to the eigenvector for

Final answer:

(a) This month’s shares:
(b) Eigenvalues and , with eigenvectors and
(d) with
(e) After months: ; long-run share:

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