Inequalites for monotonic pairs of<i>Z</i>-matrices

Smith, Ronald L.; Hu, Shu-An

doi:10.1080/03081089808818548

Cited by 10 publications

(5 citation statements)

References 7 publications

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“…In the following theorem we give some conditions on the matrices A and B so that the matrices in the set int(J) belong to the class M inv , M sing and N 0 provided det C > 0, det C = 0 and det C < 0 respectively. This generalizes Theorem 3.9, [8]. , then C 2 ∈ int(J), and det(C 2 ) = 0 so that C 2 ∈ M sing .…”

supporting

confidence: 73%

“…In the following theorem, we give a condition on the matrices A and B so that all the matrices in the interval J are M -matrices. This generalizes Theorem 3.6, [8]. Since A ∈ N 0 implies that A is irreducible it follows that C is also irreducible.…”

supporting

confidence: 71%

“…In the following theorem we give a condition on the matrices A and B so that all the matrices in the set J are N 0 -matrices. This generalizes Theorem 3.10, [8]. Corollary 2.4 (Theorem 3.10, [8]).…”

supporting

confidence: 65%

“…In the next result, we show that there are many more invertible M -matrices of which aA + bB is just one type. Our approach is much simpler than the proof of [8]. This is our main result.…”

Section: Resultsmentioning

confidence: 79%

“…In [5], Ky Fan showed that if A and B are nonsingular M -matrices with A ≤ B, then A + B is also a nonsingular M -matrix. In [8], Smith and Hu proved that if A is an M -matrix and B is a nonsingular M -matrix with A ≤ B, then aA + bB is a nonsingular M -matrix for all a, b > 0. Their proof was based on the existence of a certain semi-positive vector and the principle of mathematical induction.…”

Section: Introductionmentioning

confidence: 99%

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Intervals of certain classes of \b{Z}-matrices

Kannan

Sivakumar

2014

Discuss. Math. - General Algebra and Applications

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Let A and B be M-matrices satisfying A ≤ B and J = [A, B] be the set of all matrices C such that A ≤ C ≤ B, where the order is component wise. It is rather well known that if A is an M-matrix and B is an invertible Mmatrix and A ≤ B, then aA + bB is an invertible M-matrix for all a, b > 0. In this article, we present an elementary proof of a stronger version of this result and study corresponding results for certain other classes as well.

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supporting

confidence: 73%

supporting

confidence: 71%

supporting

confidence: 65%

“…In the next result, we show that there are many more invertible M -matrices of which aA + bB is just one type. Our approach is much simpler than the proof of [8]. This is our main result.…”

Section: Resultsmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

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Intervals of certain classes of \b{Z}-matrices

Kannan

Sivakumar

2014

Discuss. Math. - General Algebra and Applications

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Matrix Positivity

Johnson

Smith

Tsatsomeros

2020

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This encyclopedic book gives a detailed overview of our current (2019) knowledge of positive matrices and related concepts in the theory of matrix positivity. Its contents cover 208 pages and an extensive bibliography of 16 pages counts over 320 items from 1912 to 2019. The incipit of all modern studies in this field is the Perron-Frobenius theorem (1907)(1908).The book starts with a list of 53 subject specific symbols and a list of acronyms for 40 matrix classes. Chapter 1 briefly states a few mathematical concepts such as matrices, vectors, convexity, Helly's theorem, half-spaces and cones. Chapter 2 defines eleven specific positive-like matrix classes with some of their subclasses and acronyms and it states a complete containment diagram for ten of these. Chapters 3-6 deal with four of these classes with more details. These chapters develop our knowledge about semipositive matrices, P-matrices, inverse M-matrices, and copositive matrices with many newly developed short and long proofs. Chapter 3 alone contains 45 proofs of theorems and corollaries in 40 pages. Chapters 4 and 5 present interesting open research questions. Section 4.6 contains a recursive algorithm for the P-problem and Section 4.7.2 an algorithm for constructing P-matrices.Throughout the book there are applications of various positive-like matrices, such the linear complementarity problem (in Chapter 4) and iterative matrix algorithms and differential equation solvers (Chapter 5). The style of writing is very concise and full of minutiae in every page. Highlights are not stressed. Almost a quarter of the references refers to works authored or coauthored by the three authors of this book. This is a very personal assessment and a beautiful representation and re-interpretation of much of the authors' extensive work on this subject. However, standard generalizations, such as generalized Cholesky methods for symmetric matrices, inertia counting, polar decompositions and product representations by positive matrices, are missing. Neither sums of positive matrices, nor matrix pencil characterizations via matrix positivity are included. The book is almost typo-free, except of a notational inconsistency for generalized binomial coefficients that appear both in (correct) braces and in bold brackets (see pages 101-102).

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Preface

Haddad¹

2020

Matrix Positivity

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Inequalites for monotonic pairs ofZ-matrices

Cited by 10 publications

References 7 publications

Intervals of certain classes of \b{Z}-matrices

Intervals of certain classes of \b{Z}-matrices

Matrix Positivity

Preface

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