Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How to prove this statement:

If $P = [p_{ij}]_{ 1 \leqslant i,j \leqslant m} \geqslant 0$ is a primitive matrix, then there exists a $k \in \mathbb{N}$ such that $ P^{k} > 0.$ Moreover $ P^{k+i}>0$ for all $ i = 1,2, \dots $

share|cite|improve this question
That depends on your definition. Usually this is the definition of a primitive matrix. What is your definition? – Dennis Gulko Jan 24 '12 at 14:46
I use the definition appeared in (Roger A. Horn and Charles A. Johnson. Matrix analysis. Cambridge University Press, 1985" page:516) which is a matrix $ P = [p_{ij}]_{1 \leqslant i,j \leqslant m} \geqslant 0$ is called primitive if it is irreducible matrix and it has a strictly dominant eigenvalue. – Zizo Jan 24 '12 at 14:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.