Check your understanding

Problem 1

Which matrices does Perron's theorem apply to?

What classification are each matrix?

A: nonnegative, later versions of Perron's theorem apply,

primitive

B: mixed, Perron's theorem does not apply

C: positive, Perron's theorem applies

D: positive, Perron's theorem applies,

stochastic

Problem 2

Set up the transition matrix for the population dynamics of Colorado’s dragons in the mountains. Every year, 80% of the dragons in the mountains move away and 12% of dragons move into the mountains.

Does Perron's theorem apply?

Yes, as the transition matrix is strictly positive.

Find the starting vector and vector representing the ratio of dragons after one year given that the ratio of the dragons in the mountains compared to dragons outside of the mountains starts at 30/120= 25%, leaving 75% not in the mountains.

Find the eigenvectors and eigenvalues then identify the Perron root.

After confirming that Q is a stochastic, n by n, positive matrix, find the final state guaranteed by Markov's Theorem.

What does the final state mean for this application?

The dragon population in Colorado will eventually be 45/120 in the mountains and 75/120 outside of the mountains.