Dedicated to the late Professors David Hawkins and Hukukane Nikaido

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1 TWO CHARACTERIZATIONS OF INVERSE-POSITIVE MATRICES: THE HAWKINS-SIMON CONDITION AND THE LE CHATELIER-BRAUN PRINCIPLE TAKAO FUJIMOTO AND RAVINDRA R RANADE Dedicated to the late Professors David Hawkins and Hukukane Nikaido Abstract It is shown that the Hawkins-Simon condition is satisfied by any real square matrix which is inverse-postive after a suitable permutation of columns or rows One more characterization of inverse-positive matrices is given concerning the Le Chatelier-Braun principle Key words Hawkins-Simon condition Inverse-positivity Le Chatelier-Braun principle AMS subject classifications 5A5 5A48 Introduction Ineconomicsaswellasothersciencestheinverse-positivity of real square matrices has been an important topic The Hawkins-Simon condition [] so called in economics requires that every principal minor be positive and they showed the condition to be necessary and sufficient for a Z-matrix(a matrix with nonpositive off-diagonal elements) to be inverse-positive One decade earlier this was used by Ostrowski [] to define an M-matrix(an inverse-positive Z-matrix) and shown to be equivalent to some of other conditions (See Berman and Plemmons [ Ch] for many equivalent conditions) Georgescu-Roegen [] argued that for a Z-matrix it is sufficient to have only leading principal minors positive which was also proved in Fiedler and Ptak [] Nikaido s two books [8] and [9] contain a proof based on mathematical induction Dasgupta [] gave another proof using an economic interpretation of indirect input In this note the Hawkins-Simon condition is defined to be the one which requires that all the leading principal minors should be positive and shall be called in this note the weak Hawkins-Simon condition(whs for short) We prove this condition WHS is necessary for a real square matrix to be inverse-positive after a suitable permutation of columns (or rows) The proof is easy and simple by use of the elimination method One more characterization of inverse-positive matrices is offered which is related to the Le Chatelier-Braun principle in thermodynamics That is each element in the inverse of leading (n ) (n ) matrix is smaller or equal to the corresponding element in the inverse of the original inverse-positive matrix Section explains our notation then in section we present our theorems and their proofs finally giving some numerical examples and remarks in the final section 4 Notation The symbol R n means the real Euclidean space of dimension n(n ) and R n + the non-negative orthant of R n A given real n n matrix A maps from R n into itself Let a ij represent the (i j) entryofa whilex R n stands Received by the editors on Accepted for publication on Handling Editor: Department of Economics University of Kagawa Takamatsu Kagawa -85 Japan (takao@eckagawa-uacjp ranade@eckagawa-uacjp)

2 T Fujimoto and R Ranade for a column vector and x i for the i-th element of x Thesymbol(A) j means the j-thcolumnofa and(a) i means the i-throwwealsousethesymbolx (i) which represents the column (n ) vector formed by deleting the i-th element from x Similarly the symbol A (i j) means the (n ) (n ) matrix obtained by deleting the i-th row and the j-th column from A LikewiseA (j) shows the n (n ) matrix obtained by deleting the j-th column from A Let(A) i (n) be the row (n ) vector formed by deleting the n-th element from (A) i and(a) (n)j be the column (n ) vector formed by deleting the n-th element from (A) j Thesymbole i R n + means a column vector whose i-th element is unity with all the remaining entries being zero The inequality signs for vector comparison are as follows: x y iff x y R n + ; x>y iff x y R n + {}; x À y iff x y int(r n +) where int(r n +)meanstheinteriorofr n + These inequality signs are used also for matrices in a similar meaning Propositions Let us first note that when a real square matrix A is inversepositive ie A > this is equivalent to the following property: Property For any b int(r n + ) the equation Ax = b has a solution x int(rn + ) Now we can prove the following theorem Theorem Let A be inverse-positive Then the WHS condition is satisfied when a suitable permutation of columns (or rows) is made Proof Because of Property above there should be at least one positive entry in the first row of A So such a column and the first column can be exchanged: we assume the two columns have been so permuted thus () a > Next we divide the first equation of the system Ax = b by a and subtract this equation side by side from the i-th(i ) equation after multiplying this by a i to obtain a /a a n /a a a a /a a n a n a /a a n a a n /a a nn a n a n /a x x x n = In short the (i j)-entry of the modified matrix is for i j a a j a i a ij a b /a b b a /a b n b a n /a Again because of Property the second row has at least one positive entry assuming b was chosen so large to make b b a /a > We suppose the column containing this element and the second column have been permutated making

3 Inverse-Positive Matrices () a a > a The same method of elimination is then applied to the remaining (n ) equations to obtain the following: a /a α α n α α n α α n α nn x x x n = b /a β β β n where α = a a a a a a a a a a a b b a a b and β = a a and for i j > α ij = a a a j a j a i a i a ij a a a a b b a i a i b i and β i = a a This is because α = a a a a a a a Á a a a a and the other elements can be derived in a similar way When b i (i ) s are large enough β i (i ) s are all positive because a a > Then thanks to Property there exists at least one α j (j ) which is positive and we assume the column having this entry and the third column have been exchanged Hence

4 4 T Fujimoto and R Ranade () α = a a a a a a a a a > We can proceed in a similar way eliminating x 4 x n andfinally we have (4) A(nn)x n = A (n) b A(nn) From () () () and the subsequent equations we know that all the leading principal minors up to A (nn) are positive and thus when b n is large enough the RHS of (4) becomes positive Then by Property it follows > Therefore our theorem is proved for a permutation of columns For rows we can transpose a given matrix and the same proof applies Corollary When A is a Z-matrix the WHS is necessary and sufficient for A to be inverse-positive Proof Since the necessity is proved in Theorem we show the sufficiency Let us consider the elimination method used in the proof of Theorem We assume that b À When A is a Z-matrix it is easy to notice that as elimination proceeds a positive entry is always given at the upper left corner with the other entries (or coefficients) on the top equation being all non-positive while the RHS of each equation always remains positive So finallywereachtheequationofasinglevariablex n with the two coefficients on both sides being positive Thus x n > Now moving backward we find x À Since b is arbitrary this shows the WHS is sufficient Next we present a theorem which is related to the Le Chatelier-Braun principle (See Fujimoto [4]) Theorem Suppose that A is inverse-positive and the WHS is satisfied Then each element of the inverse of A (nn) is less than or equal to the corresponding element of the inverse of A Proof Thefirst column of the inverse of A can be obtained as a solution vector x R n + to the system of equations Ax = e whilethefirst column of the inverse of A (nn) as a solution vector y R+ n to the system A (nn) y = e (n) Adding these two systems with some manipulations we get the following system: (5) A x + y x n + y n x n = d (A) n (n) y

5 Inverse-Positive Matrices 5 By the Cramer s rule it follows that A (n) d A (nn) x n = =x n + (A) n (n) y Thus if x n =(A ) n > then (A) n (n) y< and if x n =then(a) n (n) y = because A (nn) > thanks to the WHS condition For the i-th(i <n) equation of (5) the Cramer s rule gives us A (ni) x i + y i =x i + (A) n (n) y From this we have y i = x i +(A ) in (A) n (n) y Therefore we can assert ½ yi <x i when (A ) n > and(a ) in > y i = x i when (A ) n = or (A ) in = For other columns we can proceed in a similar way by use of e i Corollary 4 Suppose that the inverse of A has its last column and the bottom row non-negative and (A ) nn > Then each element of the inverse of A (nn) is less than or equal to the corresponding element of the inverse of A Proof Exactly the same proof as that of Theorem can be applied because A (nn) =(A ) nn 4 Numerical Examples and Remarks The first example is a simple M- matrix and given as A and A = By exchanging two columns we have and its inverse is This satisfies the normal Hawkins-Simon condition The inverse of () is () and the entry is smaller than thus verifying Theorem ThenextoneisnotanM-matrix: A The inverse of A () is calculated as and A =

6 T Fujimoto and R Ranade = The elements (A ) (A ) and(a ) are all equal to (A () ) (A () ) and (A () ) because (A ) =and(a ) = Theentry(A () ) is however and is smaller than the corresponding entry (A ) = again confirming Theorem Thethirdexampleis A For this matrix A (44) = (A (44) ) () = These verifies Theorem (Note that On the other hand A(44) = (A(44) ) () = A = which confirms Theorem The final example is to verify Corollary 4: A and A = Since = > and => = 4 < ) 8 = 4 9 and = =

7 Inverse-Positive Matrices these equations conforms to Corollary 4 Remark 4 The Le Chatelier-Braun principle in thermodynamics states that when an equilibrium in a closed system is perturbed directly or indirectly the equilibrium shifts in the direction which can attenuate the perturbation As is explained in Fujimoto [4] the system of equations Ax = b can be solved as an optimization problem when A is an M-matrix Thus a solution x to the system can be viewed as a sort of equilibrium A similar argument can be made when A is inverse-positive In our case a perturbation is a new constraint that the n-th variable x n should be kept constant even after the vector b shifts destroying the n-th equation The changes in other variables may become smaller when the increase of those variables requires x n to be greater This is obvious in the case of an M-matrix Whatwehaveshownis that it is also the case with an inverse-positive matrix or even with a matrix with positively bordered inverse as can be seen from Corollary 4 Remark 4 Much more can be said about sensitivity analysis for the case of M-matrices We can also deal with the effects of changes in the elements of A on the solution vector x See FujimotoHerreroand Villar[5] REFERENCES [] Abraham Berman and Robert J Plemmons Nonnegative Matrices in the Mathematical Sciences Academic Press New York 99 [] Dipankar Dasgupta Using the correct economic interpretation to prove the Hawkins-Simon- Nikaido theorem: one more note Journal of Macroeconomics 4:55 99 [] Miroslav Fiedler and Vlastimil Ptak On Matrices with nonpositive off-diagonal elements and positive principal minors Czechoslovak Mathematical Journal :8 4 9 [4] Takao Fujimoto Global strong Le Chatelier-Samuelson principle Econometrica 48: 4 98 [5] Takao Fujimoto Carmen Herrero and Antonio Villar A sensitivity analysis for linear systems involving M-matrices and its application to the Leontief model Linear Algebra and Its Applications 4: [] Nicholas Georgescu-Roegen Some properties of a generalized Leontief model in Tjalling Koopmans(ed) Activity Analysis of Allocation and Production John Wiley & Sons New York 5 95 [] David Hawkins and Herbert A Simon Note: Some Conditions of Macroeconomic Stability Econometrica : [8] Hukukane Nikaido Convex Structures and Economic Theory Academic Press New York 9 [9] Hukukane Nikaido Introduction to Sets and Mappings in Modern Economics Academic Press New York 9 [] Alexander Ostrowski Über die Determinanten mit überwiegender Hauptdiagonale Commentarii Mathematici Helvetici :9 9 9

A Generalization of Theorems on Inverse-Positive Matrices

A Generalization of Theorems on Inverse-Positive Matrices Takao Fujimoto**, José A. Silva and Antonio Villar ** Department of Economic Analysis, University of Alicante, 03071, Spain Address for correspondence: