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Two characterizations of inverse-positive matrices: the Hawkins-Simon condition and the Le Chatelier-Braun principle. (English) Zbl 1069.15020
N. Georgescu-Roegen [Activity Analysis of Produciton and Allocation, Chap. X, 165–173 (1951; Zbl 0045.09607)] showed that for a Z-matrix the weak Hawkins-Simon condition, that all the leading (upper left corner) principal minors are positive [cf. D. Hawkins and H. A. Simon, Econometrica, Chicago 17, 245–248 (1949; Zbl 0036.10001)], is sufficient for inverse-positivity. The first theorem says that it is necessary for inverse positivity of a general real square matrix, up to permutations of rows or columns. It is a corollary (which is previously known) that for Z-matrices the weak Hawkins-Simon condition is necessary and sufficient for inverse-positivity. The second main theorem states, if \(A^{-1}\) has its last column and bottom row nonnegative and \(| A_{(n,n)}| >0\) then \(\forall i,j<n, (A_{(n,n)}^{-1})_{ij}\leq a_{ij}\).

MSC:
15B48 Positive matrices and their generalizations; cones of matrices
15A15 Determinants, permanents, traces, other special matrix functions
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