Exercise 12, § 6

Let $A$ be a real matrix of type $(I,I)$, irreducible and positive. For every nonzero $x\in \mathbf{R}^I$, put

$r=$ inf$_{xxi_i\in\not=0I}\frac{(Ax)_i}{x_i}$,

where ($Ax$)$_i=\sum_{k\in I}a_{ik}x_k$ is the coordinate of $Ax$ of index $i$.

a) If $x\succcurlyeq 0$ and $x\not = 0$, then $r_x$ is the greatest real number $\varrho$ such that $\varrho x\preccurlyeq Ax$. b) Suppose that $x\succcurlyeq 0$ and put $y= (1_I+$ A$)x$. Then $r_x\leqslant r_y$.

c) The restriction to $(\mathbf{R}_+^*)^I$ of the function $x\mapsto r_x$ is continuous.

d) Deduce that there exists $z\in \mathbf{R}^I$ such that $z\succcurlyeq 0$ and such that $r_z=$ sup$_{x\succcurlyeq 0}r_x$. (Use the compact sets $M =\{x\in \mathbf{R}_+^I|\sum_{i\in I}x_i= 1\}$ as well as $(1_I+$ A)$^{n-1}$(M), then Exercise 10.)

We note $r=r_z$. An element $z$ of $\mathbf{R}^I$ such that $z\succcurlyeq 0$ and $r_z=r$ will be called extremal.

e) One has $r >0$.

f) If $z$ is an extremal vector, then $Az=rz$. (If $u=Az-rz\succcurlyeq 0$ were nonzero, then $(1_I+$ A)$^{n-1}u$ would be $\succ 0$ according to Exercise 10 and, for $x= (1_I+$ A$)^{n-1}z$, one would have $Ax\succ rx.)$

g) Every extremal element is $\succ 0$.

For every vector $y$ in $\mathbf{C}^I$, we denote by $y^+$ the vector $(|y_i|)_{i\in I}$; thus one has $y^+\in \mathbf{R}^I$ and $y^+\succcurlyeq 0$.

h) Let $\alpha \in \mathbf{C}$ be an eigenvalue of A, viewed as a complex matrix, and let $y$ in $\mathbf{C}^I$ be a corresponding eigenvector. One has $|\alpha |y^+\preccurlyeq Ay^+$ and $|\alpha |\leqslant r$.

i) Deduce that if $\alpha =r$, all the coordinates of $y$ are nonzero, then that the eigenspace (in $\mathbf{C}^I$) of A associated with the eigenvalue $r$ has dimension 1.

We denote by Δ(X) the determinant of $X\cdot 1_I-A$ and by B(X) the transpose of the matrix of the cofactors of $X\cdot 1_I-A$. We denote B(X) = ($B_{ij}(X)$)$_{i,j\in I}$. We therefore have B$(X)(X\cdot 1_I-A) = (X\cdot 1_I-A$)B$(X) = \Delta (X)\cdot 1_E$ (A, III, p. 99, prop. 12). j) For every $i\in I$, the column of index $i$ of B$(r)$ is nonzero. (If this is not the case, construct an eigenvector of A associated with the eigenvalue $r$ and whose coefficient of index $i$ is zero, in contradiction with question f).) k) All the coefficients of B$(r)$ have the same sign. (First prove, using the relation $(r1_I-A$)B$(r) = 0$, that for every $i\in I$, all the coefficients of the column of index $i$ of B$(r)$ are nonzero and have the same sign, then reason with the transposes.)

l) We have $\Delta '(r)>0$ and $B_{ij}(r)>0$ for every $(i, j)\in I^2$. (Use the formula $\Delta '(X) =\sum_iB_{ii}(X)$ and observe that $\Delta '$ does not change sign on the interval $[r,+\infty ].$)

m) Conclude that $r$ is a simple root of the characteristic polynomial of A.