Qualitative Economics and Morishima Matrices

Bassett, Lowell; Habib-Agahi, Hamid; Quirk, J. P.

doi:10.2307/1909110

Cited by 26 publications

(7 citation statements)

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“…We conclude this section with the definition of Morishima and anti-Morishima matrices, as it appears in [5]: Let B be an anti-Morishima matrix. We say that B is of type i if all its diagonal entries are nonpositive, and the minimum among the orders of −B 11 and −B 22 (as they appear in definition 1) is equal to i.…”

Section: Known Results and Definitionsmentioning

confidence: 99%

“…What about k = 3? By Theorem 5.3, all extremal matrices in H SN (5,3) have six anti-Morishima 2-by-2 Schur complements of type 0. Moreover, similarly to our main result, this structure has a beautiful visualization through polyhedra.…”

Section: ) Be a Nonsingular Matrix Let B Be A Principal Submatrix Ofmentioning

confidence: 94%

“…Moreover, similarly to our main result, this structure has a beautiful visualization through polyhedra. Note that a 2-by-2 Schur complement in a 5-by-5 matrix corresponds to a set of three indices, so that if A ∈ H SN (5,3) and A/B is of order 2 then B is of order 3, and the three indices are the indices of the rows (columns) of B. For any extremal H SN(5, 3) matrix we can associate a polyhedron whose faces are triangles, such that a set of three numbers form a face if and only if the appropriate Schur complement is anti-Morishima of type 0.…”

Section: ) Be a Nonsingular Matrix Let B Be A Principal Submatrix Ofmentioning

confidence: 99%

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The structure of Schur complements in hollow, symmetric nonnegative matrices with two nonpositive eigenvalues

Farber

Johnson

2014

Linear and Multilinear Algebra

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The Schur complement structure with respect to 2-by-2 principal submatrices, in hollow, symmetric nonnegative matrices is investigated, with an emphasis on such matrices that have only two nonpositive eigenvalues. It is shown that a wide family of such Schur complements simply follows a unique and surprising structure that can be fully described in a graph theoretical language, and is predictable from the entries. For larger numbers of nonpositive eigenvalues, conjectures regarding connections to polyhedra are also presented, and proved in a special case. Relations to copositive matrices and Morishima matrices are described as well.

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Section: Known Results and Definitionsmentioning

confidence: 99%

Section: ) Be a Nonsingular Matrix Let B Be A Principal Submatrix Ofmentioning

confidence: 94%

Section: ) Be a Nonsingular Matrix Let B Be A Principal Submatrix Ofmentioning

confidence: 99%

See 1 more Smart Citation

The structure of Schur complements in hollow, symmetric nonnegative matrices with two nonpositive eigenvalues

Farber

Johnson

2014

Linear and Multilinear Algebra

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“…It will be noted that if A is a stable matrix, then A is nonsingular; we shall first summarize certain theorems concerning the class of stable qualitative matrices that have "signed inverses." THEOREM 5.9 (see [22] Proof The sufficiency of (i) is obvious while the sufficiency of (ii) follows from a similar result in [15]. To prove the necessity, assume that aiajkaki < 0 for three distinct indices i, j, k. Then the cofactor Aij of the element a in A contains two terms of opposite sign, aji x diagonal elements excluding aii and ajj and COROLLARY (see [13]).…”

mentioning

confidence: 91%

“…The following generalizes a result obtained by McKenzie for Perron-Frobenius matrices. THEOREM 5.7 (see [22] (ii) Ibhlh2l --Ibh2h3l…”

mentioning

confidence: 99%