Products of positive definite matrices. III

Ballantine, C.S.

doi:10.1016/0021-8693(68)90093-8

Cited by 27 publications

(17 citation statements)

References 2 publications

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“…If A 1 , A 2 , A 3 0 and the product A 1 A 2 A 3 is symmetric, then it is positive definite. • Result of C. Ballantine [25]. Except when A = −λI n with λ > 0 and n even, any square matrix with positive determinant can be written as the product of four positive definite matrices.…”

Section: Closedness Under Multiplicationmentioning

confidence: 99%

A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World

Hiriart-Urruty

Malick

2012

J Optim Theory Appl

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International audienceEngineering sciences and applications of mathematics show unambiguously that positive semidefiniteness of matrices is the most important generalization of non-negative real num- bers. This notion of non-negativity for matrices has been well-studied in the literature; it has been the subject of review papers and entire chapters of books. This paper reviews some of the nice, useful properties of positive (semi)definite matrices, and insists in particular on (i) characterizations of positive (semi)definiteness and (ii) the geometrical properties of the set of positive semidefinite matrices. Some properties that turn out to be less well-known have here a special treatment. The use of these properties in optimization, as well as various references to applications, are spread all the way through. The "raison d'être" of this paper is essentially pedagogical; it adopts the viewpoint of variational analysis, shedding new light on the topic. Important, fruitful, and subtle, the positive semidefinite world is a good place to start with this domain of applied mathematics

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Section: Closedness Under Multiplicationmentioning

confidence: 99%

A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World

Hiriart-Urruty

Malick

2012

J Optim Theory Appl

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“…Note that this is smaller than the generating number 5 (4 for nonscalar matrices) obtained from the positive definite matrices [2,11] and that the product of two positive definite matrices is not, in general, a P -matrix.…”

Section: P -Matrix Factorizationsmentioning

confidence: 83%

Matrix Classes That Generate All Matrices with Positive Determinant

Johnson¹,

Olesky²,

Driessche³

2003

SIAM J. Matrix Anal. & Appl.

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New factorization results dealing mainly with P -matrices and M -matrices are presented. It is proved that any matrix in Mn(R) with positive determinant can be written as the product of three P -matrices (compared with the classical result that five positive definite matrices may be needed). It is also proved that a matrix A with positive determinant can be stabilized via multiplication by a P -matrix if and only if A is not a diagonal matrix with all diagonal entries negative. Factorization into two P -matrices is considered and characterized for n = 2. Using elementary bidiagonal factorization results, it is shown that the nonsingular M -matrices, or the nonsingular totally nonnegative matrices, generate all matrices in Mn(R) with positive determinant. Further results on products of M -matrices and inverse M -matrices are given.

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“…Then, Y(x g ) is a symmetric positive definite matrix for all x g ∈ S. It has been shown in the proof of Theorem 1 that under Assumption 1, J(x g ) is a symmetric negative definite matrix for all x g ∈ S. It follows that −Z(x g ) is the product of two symmetric positive definite matrices, Y(x g ) and −J(x g ). By Theorem 2 in Ballantine (1968), all the eigenvalues of −Z(x g ) are real and positive for all x g ∈ S. Thus, all the eigenvalues of Z(x * g ) are real and negative, and local asymptotic stability of the Nash equilibrium is established.…”

Section: Lemma 1 B(x G ) Is Negative Semidefinite For All X G ∈ Smentioning

confidence: 89%

Warm-glow giving in networks with multiple public goods

Richefort

2018

Int J Game Theory

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This paper explores a voluntary contribution game in the presence of warm-glow effects. There are many public goods and each public good benefits a different group of players. The structure of the game induces a bipartite network structure, where players are listed on one side and the public good groups they form are listed on the other side. The main result of the paper shows the existence and uniqueness of a Nash equilibrium. The unique Nash equilibrium is also shown to be locally asymptotically stable. Then the paper provides some comparative statics analysis regarding pure redistribution, taxation and subsidies. It appears that small redistributions of wealth may sometimes be neutral, but generally, the effects of redistributive policies depend on how public good groups are related in the contribution network structure.

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Products of positive definite matrices. III

Cited by 27 publications

References 2 publications

A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World

A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World

Matrix Classes That Generate All Matrices with Positive Determinant

Warm-glow giving in networks with multiple public goods

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