Exercise 13, § 6

Let A (resp. C) be a real (resp. complex) square matrix of type $(I,I)$. We suppose that A is positive and irreducible and that $C^+\preccurlyeq A$, where, similarly to Exercise 12, we denote by $C^+$ the matrix whose coefficients are the moduli of the coefficients of C, and we write $C^+\preccurlyeq A$ to mean that the coefficients of $A-C^+$ are positive.

We use again the notations of the preceding exercise.

a) Let $\gamma \in \mathbf{C}$ be an eigenvalue of C and let $y$ in $\mathbf{C}^I$ be an associated eigenvector. We have $|\gamma |y^+\preccurlyeq C^+y^+\preccurlyeq Ay^+$ and $|\gamma |\leqslant r_{y^+}\leqslant r$.

b) We suppose in addition that $|\gamma |=r$. Then $y^+$ is an extremal vector for A (cf. question f) of Exercise 12), and we have the relations $Ay^+=C^+y^+=ry^+$ and A = $C^+$.

Let $u$ (resp. $u_j$, for $j\in I$) be the unique complex number such that $\gamma =|\gamma |u$ (resp. $y_j=|y_j|u_j$, for $j\in I$). Let D be the complex square matrix of type $(I,I)$ whose diagonal coefficients are $(u_j)_{j\in I}$ and whose other coefficients are zero, so that $y=Dy^+$. Let F $=u^{-1}D^{-1}$ CD.

c) We have $Fy^+=ry^+$ and $F^+y^+=Fy^+$. (Observe that $F^+=C^+.$)

d) We have F = $F^+$ and C $=u$DAD$^{-1}$.