I have a concern regarding primitive matrices. We learned in class several theorems to determine whether or not a matrix is primitive. I'll state them bellow for reference:
- If an irreducible non-negative matrix $A$ has $h$ eigenvalues, $\lambda_1, \lambda_2, ..., \lambda_h$ of maximum modulus $|{\lambda_1}|=|{\lambda_i}|, i=1,2,...,h$, then A is called primitive if $h=1$ and imprimitive if $h>1$
- A non-negative matrix $A$ is primitive iff some power of $A$ is positive. (i.e. $A^p>0$ for some integer $p>1$).
- An irreducible Leslie matrix is primitive iff the birth rates satisfy the following relationship (let greatest common divisor = g.c.d): $$g.c.d.\{i|b_i>0\}=1$$
So with these theorems in mind, I see this inconsistency with them, hopefully someone can point out where I'm misunderstanding one of these theorems.
Ex: If we have the following matrix $$L=\begin{bmatrix} 0 & b_2 & 0 & b_4\\ s_1 & 0 & 0 & 0\\ 0 & s_2 &0 &0 \\ 0 & 0 & s_3 & 0 \end{bmatrix}$$, with $b_i, s_i >0$.
This is an irreducible Leslie matrix, and depending on what values of $b_2$ and $b_4$, we can make it so that $g.c.d.\{i|b_i>0\}=1$, example if we set $b_2=3$, and $b_4=4$.
Okay, so by theorem 3, the matrix should be primitive. However, there's no positive integer $p$ such that $L^p>0$, which contradicts theorem 2 (I tried several different values for p, going up to 20 and none of them were strictly positive matrices). I really don't understand this. Any help would be greatly appreciated!