Exercise 22, § 6

Let A be a real square matrix of type $(I,I)$, positive and irreducible; let $\varrho$ be its spectral radius. Let $d\geqslant 1$ be the integer determined by exercise 14. We shall say that A is primitive if $d= 1$. Let Δ(X) be the polynomial det(X $\cdot 1_I-A$).

a) We write $\Delta (X) = X^n+a_1X^{n_1}+\cdots +a_tX^{n_t}$, where $a_1, . . . , a_t$ are nonzero real numbers and $n_1, . . . , n_t$ natural integers such that $n > n_1>$ $\cdots > n_t$. Exercise prove that $d$ is the greatest common divisor of $n-n_1$, $n_1-n_2, . . . , n_{t-1}-n_t$.

b) There exists a nonzero integer $m$ and a polynomial $g$ such that $\Delta (X) =g(X^d)X^m$.

c) If there exists an integer $p >0$ such that $A^p$ is strictly positive, then A is primitive.

d) Suppose that A is primitive. Introduce the matrices C (X) of exercise 15 and put, with the notations of this exercise, $\mathsf{X}(X) = \Delta (X)/\Delta_1(X)$. The limit of the sequence ($A^k/\varrho^k$)$_{k\in\mathbf{N}}$ is equal to C$(\varrho )/\mathsf{X}'(\varrho )$. (One may use a decomposition of A into Jordan blocks.) Using the results of exercise 15, deduce that there exists an integer $p >0$ such that $A^p$ is strictly positive.

e) Suppose that there exists an integer $q >0$ such that $A^q$ is reducible. Let $p$ be the greatest common divisor of $q$ and $d$. Proving that there exists a partition $(I_s)_{1\leqslant s\leqslant p}$ of I into nonempty parts for which $A^q$ is block diagonal, with irreducible diagonal blocks and spectral radius $\varrho^q$.

f) If A is primitive, then $A^p$ is irreducible for every $p >0$.

g) There exists a partition $(I_s)_{1\leqslant s\leqslant d}$ of I into nonempty parts for which $A^d$ is block diagonal, with primitive diagonal blocks and spectral radius equal to $\varrho^d$.