Appendix

Characteristic roots and vectors Since, in this paper, price systems are expressed as characteristic vectors of nonnegative matrices, it is useful to state the following theorems, which have been proven in the theory of matrices: 
Definition: The characteristic root l_{0} of a square matrix A is called a maximum characteristic root if, for all the remaining characteristic roots, l_{i} of matrix A, l_{i} £ l_{0} holds true. 
Theorem (1) A nonnegative matrix A ³0 always has a nonnegative maximum characteristic root l_{0}, and a nonnegative characteristic vector X_{0}. For any arbitrarily chosen l > l_{0}, (lI  A)^{1 }³ 0; and(lI  A)^{1} /dl£ 0. Note: If l_{0 }is a maximum characteristic root of matrix A, it is also the maximum characteristic root of matrix A’.

Theorem (2) Let A ³ 0 be irreducible. Then there exists a positive maximum characteristic root l_{0} and accompanying it a positive characteristic vector X_{0}. No other positive characteristic vector of matrix A that is linearly independent of X_{0 }exists. For any arbitrarily chosen l > l_{0}, (lI  A)^{1 }> 0; and(lI  A)^{1} /dl< 0.

Theorem (3) Given two nonnegative matrices A and A* with maximum characteristic roots l_{0} and l^{*}_{0}, A <A* (A ¹ A*), then l_{0 }£_{ }l^{*}_{0}. If A is an irreducible matrix then l_{0 }<_{ }l^{*}_{0}.

Proof of these theorems can be found, for instance, in GANTMACHER (1966).




