Exercise 20, § 6

Let A be a real square matrix of type $(I,I)$, positive but not necessarily irreducible; let $\varrho$ be the spectral radius of A. We retain the notations B(X) and C(X) of the preceding exercises.

a) The real number $\varrho$ is an eigenvalue of A and there exists at least one eigenvector $\succcurlyeq 0$ associated with $\varrho$.

b) For every real number $\lambda \geqslant r$, the matrices B$(\lambda )$ and $B'(\lambda )$ are positive. (For the second matrix, one may proceed by induction on the cardinal of I.)

c) The matrix C$(\lambda )$ is positive for every $\lambda \geqslant r$. d) If A is irreducible, then for every real number $\lambda > r$, the matrices B$(\lambda )$, C$(\lambda )$ and $(\lambda 1_I-A)^{-1}$ are strictly positive.

e) Let J be a subset of I different from I. Let $\varrho_J$ be the spectral radius of the submatrix $(a_{i,j})_{i,j\in J}$. One has $\varrho_J\leqslant \varrho$, and this inequality is strict if A is irreducible. Conversely, if A is reducible, there exists a subset $J\subset I$, different from I, such that $\varrho_J=\varrho$.

f) Let J be a subset of I different from I and such that the spectral radius of the submatrix $(a_{i,j})_{i,j\in J}$ is equal to $\varrho$. For every subset K such that $J\subset K\subset I$, the spectral radius of the matrix $(a_{i,j})_{i,j\in K}$ is equal to $\varrho$. Deduce that $A\succ 0$ if and only if all the diagonal coefficients of B$(r)$ are strictly positive.

g) For every real number $\lambda > \varrho$ and for every subset J of I such that $J\not = I$, the determinant of the matrix $(\lambda \delta_{i,j}-a_{i,j})_{i,j\in J}$ is strictly positive.

h) Let $\lambda$ be a real number; suppose that the determinant of $(\lambda \delta_{i,j}-a_{i,j})_{i,j\in J}$ is strictly positive for every subset $J\subset I$ different from I. Then $\lambda > \varrho$.

In the following questions, one supposes that I is the interval $[1, n]$ of $\mathbf{N}$. i) Let $(g_{i,j})_{1\leqslant i,j\leqslant n}$ be a real matrix such that $g_{i,j}\leqslant 0$ for every $i\not =j$ and such that the determinants of the matrices $(g_{i,j})_{1\leqslant i,j\leqslant m}$ are strictly positive for every integer $m$ such that $1\leqslant m\leqslant n$. Then one has det($g_{i,j}$)$_{i,j\in J}>0$ for every subset J of I.

j) Let $\lambda$ be a real number; suppose that for every integer $m$ such that $1\leqslant m\leqslant n$, the determinant det($\lambda \delta_{i,j}-a_{i,j}$)$_{1\leqslant i,j\leqslant m}$ is strictly positive. Then $\lambda > \varrho$.

k) Let C $= (c_{i,j})_{1\leqslant i,j\leqslant n}$ be a real matrix such that $c_{i,j}\geqslant 0$ for every $i\not =j$. If $(-1)^k$ det($c_{i,j}$)$_{1\leqslant i,j\leqslant m}$ is strictly positive for every integer $m$ such that $1\leqslant m\leqslant n$, then the real parts of the eigenvalues of C (viewed as a complex matrix) are negative.