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1
Content available remote

On a nonnegative irreducible matrix that is similar to a positive matrix

100%
Loewy R.
Open Mathematics
|
2012
|
tom 10
|
nr 1
303-311
EN
Let A be an n×n irreducible nonnegative (elementwise) matrix. Borobia and Moro raised the following question: Suppose that every diagonal of A contains a positive entry. Is A similar to a positive matrix? We give an affirmative answer in the case n = 4.
2
Content available remote

Singular M-matrices which may not have a nonnegative generalized inverse

81%
Sushama A. N., Premakumari K., Sivakumar K. C.
Special Matrices
|
2014
|
tom 2
|
nr 1
EN
A matrix A ∈ ℝn×n is a GM-matrix if A = sI − B, where 0 < ρ(B) ≤ s and B ∈WPFn i.e., both B and Bt have ρ(B) as their eigenvalues and their corresponding eigenvector is entry wise nonnegative. In this article, we consider a generalization of a subclass of GM-matrices having a nonnegative core nilpotent decomposition and prove a characterization result for such matrices. Also, we study various notions of splitting of matrices from this new class and obtain sufficient conditions for their convergence.
3
Content available remote

A Hadamard product involving inverse-positive matrices

81%
Gassó M. T., Torregrosa J. R., Abad M.
Special Matrices
|
2015
|
tom 3
|
nr 1
EN
In this paperwe study the Hadamard product of inverse-positive matrices.We observe that this class of matrices is not closed under the Hadamard product, but we show that for a particular sign pattern of the inverse-positive matrices A and B, the Hadamard product A ◦ B−1 is again an inverse-positive matrix.
4
Content available remote

A note on regularity and positive definiteness of interval matrices

81%
Farhadsefat R., Lotfi T., Rohn J.
Open Mathematics
|
2012
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tom 10
|
nr 1
322-328
EN
We present a sufficient regularity condition for interval matrices which generalizes two previously known ones. It is formulated in terms of positive definiteness of a certain point matrix, and can also be used for checking positive definiteness of interval matrices. Comparing it with Beeck’s strong regularity condition, we show by counterexamples that none of the two conditions is more general than the other one.
5
Content available remote

A counterexample to the Drury permanent conjecture

81%
Hutchinson G.
Special Matrices
|
2017
|
tom 5
|
nr 1
301-302
EN
We offer a counterexample to a conjecture concerning the permanent of positive semidefinite matrices. The counterexample is a 4 × 4 complex correlation matrix.
6
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The 123 theorem of Probability Theory and Copositive Matrices

62%
Kovačec A., Moreira M. M. R., Martins D. P.
Special Matrices
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2014
|
tom 2
|
nr 1
EN
Alon and Yuster give for independent identically distributed real or vector valued random variables X, Y combinatorially proved estimates of the form Prob(∥X − Y∥ ≤ b) ≤ c Prob(∥X − Y∥ ≤ a). We derive these using copositive matrices instead. By the same method we also give estimates for the real valued case, involving X + Y and X − Y, due to Siegmund-Schultze and von Weizsäcker as generalized by Dong, Li and Li. Furthermore, we formulate a version of the above inequalities as an integral inequality for monotone functions.
7
Content available remote

Extensions of Three Matrix Inequalities to Semisimple Lie Groups

62%
Liu X., Tam T.-Y.
Special Matrices
|
2014
|
tom 2
|
nr 1
EN
We give extensions of inequalities of Araki-Lieb-Thirring, Audenaert, and Simon, in the context of semisimple Lie groups.
8
Content available remote

The higher rank numerical range of nonnegative matrices

52%
Aretaki A., Maroulas I.
Open Mathematics
|
2013
|
tom 11
|
nr 3
435-446
EN
In this article the rank-k numerical range ∧k (A) of an entrywise nonnegative matrix A is investigated. Extending the notion of elements of maximum modulus in ∧k (A), we examine their location on the complex plane. Further, an application of this theory to ∧k (L(λ)) of a Perron polynomial L(λ) is elaborated via its companion matrix C L.
9
Content available remote

Completely positive matrices over Boolean algebras and their CP-rank

52%
Mohindru P.
Special Matrices
|
2015
|
tom 3
|
nr 1
EN
Drew, Johnson and Loewy conjectured that for n ≥ 4, the CP-rank of every n × n completely positive real matrix is at most [n2/4]. In this paper, we prove this conjecture for n × n completely positive matrices over Boolean algebras (finite or infinite). In addition,we formulate various CP-rank inequalities of completely positive matrices over special semirings using semiring homomorphisms.
10
Content available remote

Inertias and ranks of some Hermitian matrix functions with applications

42%
Zhang X., Wang Q.-W., Liu X.
Open Mathematics
|
2012
|
tom 10
|
nr 1
329-351
EN
Let S be a given set consisting of some Hermitian matrices with the same size. We say that a matrix A ∈ S is maximal if A − W is positive semidefinite for every matrix W ∈ S. In this paper, we consider the maximal and minimal inertias and ranks of the Hermitian matrix function f(X,Y) = P − QXQ* − TYT*, where * means the conjugate and transpose of a matrix, P = P*, Q, T are known matrices and for X and Y Hermitian solutions to the consistent matrix equations AX =B and YC = D respectively. As applications, we derive the necessary and sufficient conditions for the existence of maximal matrices of $$H = \{ f(X,Y) = P - QXQ* - TYT* : AX = B,YC = D,X = X*, Y = Y*\} .$$ The corresponding expressions of the maximal matrices of H are presented when the existence conditions are met. In this case, we further prove the matrix function f(X,Y)is invariant under changing the pair (X,Y). Moreover, we establish necessary and sufficient conditions for the system of matrix equations $$AX = B, YC = D, QXQ* + TYT* = P$$ to have a Hermitian solution and the system of matrix equations $$AX = C, BXB* = D$$ to have a bisymmetric solution. The explicit expressions of such solutions to the systems mentioned above are also provided. In addition, we discuss the range of inertias of the matrix functions P ± QXQ* ± TYT* where X and Y are a nonnegative definite pair of solutions to some consistent matrix equations. The findings of this pape extend some known results in the literature.
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