Two characterizations of inverse-positive matrices: the Hawkins-Simon condition and the Le Chatelier-Braun principle

Fujimoto, Takao; Ranade, Ravindra R.

doi:10.13001/1081-3810.1122

Cited by 72 publications

(46 citation statements)

References 10 publications

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“…An inversepositive matrix A is a square invertible matrix that satisfies the above condition that all elements of the inverse are nonnegative. Intuitively, inverse-positive matrices play an important role in many real-world systems (Fujimoto and Ranade 2004), as production volumes in an economic system and chemical concentrations in a chemical system are necessarily non-negative. So let us consider the case of LCA.…”

Section: Discussionmentioning

confidence: 99%

Rigorous proof of fuzzy error propagation with matrix-based LCI

Heijungs

Tan

2010

Int J Life Cycle Assess

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Background, aim, and scope Propagation of parametric uncertainty in life cycle inventory (LCI) models is usually performed based on probabilistic Monte Carlo techniques. However, alternative approaches using interval or fuzzy numbers have been proposed based on the argument that these provide a better reflection of epistemological uncertainties inherent in some process data. Recent progress has been made to integrate fuzzy arithmetic into matrix-based LCI using decomposition into α-cut intervals. However, the proposed technique implicitly assumes that the lower bounds of the technology matrix elements give the highest inventory results, and vice versa, without providing rigorous proof. Materials and methods This paper provides formal proof of the validity of the assumptions made in that paper using a formula derived in 1950. It is shown that an increase in the numerical value of an element of the technology matrix A results in a decrease of the corresponding element of the inverted matrix A -1 , provided that the latter is non-negative. Results It thus follows that the assumption used in fuzzy uncertainty propagation using matrix-based LCI is valid when A -1 does not contain negative elements.Discussion In practice, this condition is satisfied by feasible life cycle systems whose component processes have positive scaling factors. However, when avoided processes are used in order to account for the presence of multifunctional processes, this condition will be violated. We then provide some guidelines to ensure that the necessary conditions for fuzzy propagation are met by an LCI model. Conclusions The arguments presented here thus provide rigorous proof that the algorithm developed for fuzzy matrix-based LCI is valid under specified conditions, namely when the inverse of the technology matrix is non-negative. Recommendations and perspectives This paper thus gives the conditions for which computationally efficient propagation of uncertainties in fuzzy LCI models is strictly valid.Keywords Fuzzy uncertainty propagation . Matrix-based LCA . Uncertainty 1 Background, aim, and scopeIn a seminal paper on uncertainty analysis in life cycle assessment (LCA), Heijungs (1996) discussed the issue of combining lower and upper values of LCA parameters to obtain an idea of the uncertainty range of LCA results. In fact, it was concluded that this was not a feasible approach for LCA systems of a sufficiently large size. The argument was that it cannot be predicted a priori if for a certain LCA parameter the lower or the upper value will produce the lowest LCA result. "This would imply that all combinations of upper and lower values must be tried in order to find the upper and lower values of the result. If there are 10,000 figures used as input data-a typical number for a mediocre LCI-the number of combinations is 2 10000 , a number which

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Section: Discussionmentioning

confidence: 99%

Rigorous proof of fuzzy error propagation with matrix-based LCI

Heijungs

Tan

2010

Int J Life Cycle Assess

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“…The main properties of our technique are consequences of the theory of M-matrices [Fujimoto and Ranade 2004], which are nonsingular, square matrices with the property that their inverses have only positive entries.…”

Section: Total Positivity Conjecturementioning

confidence: 99%

“…The black circle represents the known approximation to the exact solutions at the time t k , and the crosses denote the unknown, new approximations at the time t k+1 . serves the boundedness and the positivity of the solutions of (1), and it makes use of the nonsingularity properties of M-matrices [Fujimoto and Ranade 2004]. Proposition 1.…”

Section: Total Positivity Conjecturementioning

confidence: 99%

Untitled

2012

Involve

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“…This section and the next are devoted to the study of a class of matrices introduced by Fujimoto and Ranade [6]. 2).…”

Section: Whs After Reordering Of Columnsmentioning

confidence: 99%

“…With no assumption on the signs of the off-diagonal coefficients, three characterizations of the WHS property are given (section 3). Fujimoto and Ranade [6] have recently considered matrices which are of the WHS type after a suitable reordering of columns (these matrices are said to be of the FR type) and shown that an inverse-semipositive matrix has this property. This result is generalized and we show that, since the FR family of matrices is invariant by a group of transforms, the identification of the FR matrices should take into account the associated group (section 4).…”

Section: Introductionmentioning

confidence: 99%

The weak Hawkins-Simon Condition

Bidard¹

2007

ELA

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Abstract.A real square matrix satisfies the weak Hawkins-Simon condition if its leading principal minors are positive (the condition was first studied by the French mathematician Maurice Potron). Three characterizations are given. Simple sufficient conditions ensure that the condition holds after a suitable reordering of columns. A full characterization of this set of matrices should take into account the group of transforms which leave it invariant. A simple algorithm able, in some cases, to implement a suitable permutation of columns is also studied. The nonsingular Stiemke matrices satisfy the WHS condition after reorderings of both rows and columns.

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Two characterizations of inverse-positive matrices: the Hawkins-Simon condition and the Le Chatelier-Braun principle

Cited by 72 publications

References 10 publications

Rigorous proof of fuzzy error propagation with matrix-based LCI

Rigorous proof of fuzzy error propagation with matrix-based LCI

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The weak Hawkins-Simon Condition

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