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Thus cross multiplying by exp ( - l-lt), lexp ( B t ) - exp (zt)wvlI I exp [t(S - p)]Y where 6 - p < r - p = z. Hence choosing z' to be any fixed number in the interval 6 - p < z' < z, the assertion follows. 0 Perron Matrices Irreducible non-negative matrices T, and matrices B and - B where B is an irreducible ML-matrix are special classes of the set of Perron matrices. 2. An (n x n ) matrix A = {aij)is said to be a Perron matrix if f ( A ) > 0 for some polynomial f with real coefficients. 3 Some Extensions of the Perron-Frobenius Structure Thus if T is non-negative irreducible, an appropriate polynomial,J is given by since the matrix n f ( T )= 1T i i=l will have a strictly positive contribution for any one of its elements from at least one of the T i , i = 1, .

1. Corollary 1. r I max C aij with a similar result for the columns. We proved in the course of the development, that min x r _< max aij j i j and implicitly (through considering min j 7') 1 aij i that C aij I z I max 1a i j . i i j Corollary 2. Either Adj (TI- A ) > 0 or -Adj (zl - A ) > 0. (Proved in the course of (c)). Corollary 3 (Subinvariance). e. sc'x - zc'x = = sx. 20. Then Hence, since s - z i f and only i f Ax - C'AX= c:f, = clf: df'/clx, the result follows. 0 Since the class of Perron matrices includes (irreducible) periodic nonnegative matrices, it follows that there may be other eigenvalues of A which have modulus equal to that of z.

Mathematically) identical with the Leontief set-up. Let the elements of x , denote the output at time stage k of various industries as before; let P, 0< I 1 be the proportion (the same for each industry) of output at any stage available for internal use, a proportion 1 - P being needed to meet external demand; let tij be the amount of output of industry i per unit of input of industry j, at the next time stage, and let c 2 0, # 0, be an 'external input' vector, the same at all time stages. l) of the 'stationary system'.

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