Exercise 21, § 6

Let A be a real square matrix of type $(I,I)$, positive; let $\varrho$ be its spectral radius.

(a) Prove that there exist integers $p$ and $q$ such that $0\leqslant q\leqslant p$ and a partition $(I_s)_{1\leqslant s\leqslant p}$ of I into non-empty parts such that the corresponding block decomposition A = ($A_{s,t}$)$_{1\leqslant s,t\leqslant p}$ has the following property:

– The matrices $A_{s,s}$ are irreducible ;

– We have $A_{s,t}= 0$ if $t > s$;

– We have $A_{s,t}= 0$ if $1\leqslant t < s\leqslant q$;

– For every integer $s$ such that $q\leqslant s\leqslant p$, there exists $t$ such that $A_{s,t}\not = 0$.

(b) Determine all the possibilities for such a partition. (Prove in particular that the integers $q$ and $p$ are uniquely determined, analogously the sets $\{I_1, . . . ,I_g\}$ and the sets $\{I_{q+1}, . . . ,I_p\}.$)

(c) There exists an eigenvector $z\succ 0$ of A associated with the eigenvalue $\varrho$ if and only if, for every integer $s\in [1, p]$, the spectral radius of $A_{s,s}$ is equal to $\varrho$ if $s\leqslant q$ and is strictly lower than $\varrho$ otherwise.

(d) Conclude that in order that A and $^tA$ both admit a strictly positive extremal vector, it is necessary and sufficient that there exist a partition as above such that $A_{s,t}= 0$ for $1\leqslant t < s$ and $q < s\leqslant p$, and such that the matrices $A_{s,s}$ for $s\leqslant q$ have a spectral radius equal to $\varrho$.

(e) If A and $^tA$ both admit a strictly positive extremal vector, and if $\varrho$ has spectral multiplicity equal to 1, then A is irreducible.