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Prices 

Sekerka - Kyn - Hejl: Price Systems Computable from Input-Output Coefficients

 

 

Appendix

 

 

Characteristic roots and vectors

Since, in this paper, price systems are expressed as characteristic vectors of non-negative matrices, it is useful to state the following theorems, which have been proven in the theory of matrices:

Definition: The characteristic root l0 of a square matrix A is called a maximum characteristic root if, for all the remaining characteristic roots, li of matrix A,  |li  |l0|   holds true.

Theorem (1)

A non-negative matrix A 0 always has a non-negative maximum characteristic root l0, and a non-negative characteristic vector X0.

For any arbitrarily chosen l > l0,  (lI - A)-1 0;   and(lI - A)-1 /dl 0.

Note: If  l0 is a maximum characteristic root of matrix A, it is also the maximum characteristic root of matrix A.

 

Theorem (2)

Let A 0 be irreducible. Then there exists a positive maximum characteristic root l0 and accompanying it a positive characteristic vector X0. No other positive characteristic vector of matrix A that is linearly independent of X0 exists.

For any arbitrarily chosen l > l0,  (lI - A)-1 > 0;   and(lI - A)-1 /dl< 0.

 

 

Theorem (3)

Given two non-negative matrices A and A* with maximum characteristic roots l0 and l*0, A <A* (A A*), then  l0 l*0. If A is an irreducible matrix then l0 < l*0

 

 

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

 

 

 

 

 

 

 

 

 

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