We now state a set of useful theorems with proofs omitted.

Theorem: (Fundamental Theorem of Algebra) an algebraic expression of the polynomial form:

a

_{nl }^{n}+ a_{n-1l }^{n-1}_{ }+ .... + a_{2l }^{2}+ a_{1l }+ a_{0}= 0where a

_{0}ｹ 0, has exactly n complex or real roots l_{1}, ..., l_{n }(repetition is, of course, possible).

Theorem: If the coefficients of a polynomial (a_{1}, a_{2}, etc.) are real, then any complex roots will come in conjugate pairs, i.e. l = u ｱ iv where i is the imaginary operator i =ﾖ -1 and u and v are real numbers, so that u is the "real part" of l (i.e., Re(l ) = u) and v the "imaginary part" of l (i.e. Im(l ) = v).

Theorem: (Descartes's Rule of Signs) Let a_{0}, ..., a_{n}be the sequence of coefficients in a polynomial and let k be the total number of "changes of sign" from one coefficient to the next in that sequence. Then the number of positive real roots of the polynomial is equal to k minus a positive even number. (note: if k = 1, then there is exactly one positive real root).

We now turn to the Perron-Frobenius theorems on non-negative square matrices. They are of two types: one for the general case (I), the other for the irreducible case (II).

Theorem: (Perron-Frobenius I) IfAis a non-negative square matrix then the following hold:(a)

Ahas a non-negative real eigenvalue(b) no real eigenvalue of

Acan have an absolute value larger than the largest real eigenvalue l_{m}(c) at least one right eigenvector and one left eigenvector associated with l

_{m}are semipositive.(d) [l

I-A]^{-1}>> 0 for each l > l_{m}(e) l

_{m}is a nondecreasing function of each of the elements of matrixA.

Theorem: (Perron-Frobenius II) IfAis anirreduciblenon-negative square matrix then the following hold:(a)

Ahas a positive maximum eigenvalue l_{m}(b) the right eigenvector

xassociated with l_{m}is positive, i.e. forAx= l_{m}x,x> 0.(c) the left eigenvector

x｢associated with l_{m}is also positive, i.e.x｢A=x｢l_{m},x｢> 0.(d) if l is any eigenvalue of

A, then |l | ｣ l_{m}(e) l

_{m}is a continuous increasing function of the elements inA.(f) the maximum eigenvalue for any submatrix of

Ais smaller than the maximum eigenvalue forA.(g) To each real eigenvalue l of

Adifferent from l_{m}, there corresponds an eigenvectorxｹ0such which has at least one negative component.(h) Given a real number m = (1/n ) > 0, if m > l

_{m}(thus n < 1/l_{m}) then:

(m I - A)^{-1}> 0

(I - n A)^{-1}> 0(i) max

_{i}a^{i}1ｳ l_{m}ｳ min_{i}a^{i}1wherea^{i}is the ith row ofA. (i.e. maximum eigenvalue lies between the maximum and minimum of row sums ofA).

Proofs of the various aspects of the Perron-Frobenius theorems (plus extensions) are given in Debreu and Herstein (1953), Morishima (1964), Murata (1977), Nikaido (1960), Pasinetti (1975), Takayama (1974) and Kurz and Salvadori (1995).

Now we add the following definition from Hawkins and Simon (1949) and Georgescu-Roegen (1951):

Hawkins-Simon: a matrixAis "productive" or fulfills the "Hawkins-Simon conditions" if all the principal leading minors ofAare positive.

and consequently another pair of theorems:

Theorem: LetAbe an n ｴ n matrix where a_{ij}｣ 0 for i ｹ j, then there is anxｳ0such thatAx>0iffAfulfills the Hawkins-Simon condition.

Theorem: LetAbe an n ｴ n matrix where a_{ij}｣ 0 for i ｹ j, then for anycｳ0, there exists anxｳ0such thatAx=ciffAfulfills the Hawkins-Simon conditions.

Proofs of the above propositions can be found in Nikaido (1960; 1968: p.92).

Theorem: (Routh-Hurwitz) A necessary and sufficient condition that all the roots of the n-degree polynomial equation with real coefficients:

a

_{n}l_{ }^{n}+ a_{n-1}l_{ }^{n-1}_{ }+ .... + a_{2}l^{2}+ a_{1}l+ a_{0}= 0to have negative real parts is that all the principal leading minors of the following matrix are strictly positive:

a

_{1}a_{0}0 0 ... ...a

_{3}a_{2}a_{1}a_{0}... ...a

_{5}a_{4}a_{3}a_{2}... ...... ... ... ... ... ...

0 0 0 0 ... a

_{n}

Note that the structure of the matrix for the Routh-Hurwitz conditions. The matrix is
obtained as follows. The coefficients of the polynomial from a_{1} to a_{n}
are written out on the main diagonal. The columns consist in turn of coefficients with
only odd or even subscripts, ith the coefficient a_{0} included among the latter.
All the other entries of the matrix corresponding to coefficients with subscripts greater
than n or less than 0 are set equal to 0.

From the Routh-Hurwitz conditions, it is immediately obvious that when we are
coinsidering the characteristic equation of a 2 ｴ 2 matrix **A**:

l

^{2}- (trA)l+ |A| = 0

that a necessary and sufficient condition for the real parts of all eigenvalues to be
negative is that |**A**| > 0 and tr**A** < 0. A computationally simpler
form of the Routh-Hurwitz conditions are the "Modified Routh-Hurwitz" conditions
given in Murata (1977: p.92).

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