Positive Definite Matrices and Sylvester's Criterion

Gilber, George T.

doi:10.2307/2324036

Cited by 112 publications

(30 citation statements)

References 1 publication

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“…Since we do not need the eigenvalues themselves -only their signs -this constraint can also be derived by showing that the matrix is positive definite (so that it has only positive eigenvalues). From Sylvester's Criterion, a real-symmetric matrix is positive definite if and only if all its leading principal minors are positive (Gilbert 1991). Applying this criterion to the matrix A leads to the result (52).…”

Section: Stability Of the Critical State: The Second Variationmentioning

confidence: 99%

Energy optimization in binary star systems: explanation for equal mass members in close orbits

Adams

Batygin

Bloch

2020

Monthly Notices of the Royal Astronomical Society

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Observations indicate that members of close stellar binaries often have mass ratios close to unity, while longer-period systems exhibit a more uniform massratio distribution. This paper provides a theoretical explanation for this finding by determining the tidal equilibrium states for binary star systems -subject to the constraints of conservation of angular momentum and constant total mass. This work generalizes previous treatments by including the mass fraction as a variable in the optimization problem. The results show that the lowest energy state accessible to the system corresponds to equal mass stars on a circular orbit, where the stellar spin angular velocities are both synchronized and aligned with the orbit. These features are roughly consistent with observed properties of close binary systems. We also find the conditions required for this minimum energy state to exist: [1] The total angular momentum must exceed a critical value, [2] the orbital angular momentum must be three times greater than the total spin angular momentum, and [3] the semimajor axis is bounded from above. The last condition implies that sufficiently wide binaries are not optimized with equal mass stars, where the limiting binary separation occurs near a 0 ≈ 16R * .

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Section: Stability Of the Critical State: The Second Variationmentioning

confidence: 99%

Energy optimization in binary star systems: explanation for equal mass members in close orbits

Adams

Batygin

Bloch

2020

Monthly Notices of the Royal Astronomical Society

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“…Sylvester's criterion states that a Hermitian matrix M is positive definite if and only if the leading principal minors are positive [38]. For the matrix B 1 , the three principal minors are Similarly, for the matrix B 2 , the three principal minors are In order to see a set of global necessary conditions for positivity at arbitrary k = 0, we consider first large and small values of k = 0 separately.…”

Section: (C) Necessary and Sufficient Conditions For Real Wave Propagmentioning

confidence: 99%

Real wave propagation in the isotropic-relaxed micromorphic model

et al. 2017

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For the recently introduced isotropic-relaxed micromorphic generalized continuum model, we show that, under the assumption of positive-definite energy, planar harmonic waves have real velocity. We also obtain a necessary and sufficient condition for real wave velocity which is weaker than the positive definiteness of the energy. Connections to isotropic linear elasticity and micropolar elasticity are established. Notably, we show that strong ellipticity does not imply real wave velocity in micropolar elasticity, whereas it does in isotropic linear elasticity.

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“…Some proofs of the Sylvester criterion (see, e. g., Gilbert (1991), Thrall and Tornheim (1957)) are based on "geometric" properties of linear spaces; these proofs are in general quite concise, however not too suitable for a course of Linear Algebra for undergraduates. The elegant proof of Gilbert (1991) can be schematized as follows.…”

Section: F) Proofs Based On Properties Of Linear Spacesmentioning

confidence: 99%

“…The elegant proof of Gilbert (1991) can be schematized as follows. As already noted before, the necessity part of the proof of the Sylvester criterion shows no particular difficulty.…”

Section: F) Proofs Based On Properties Of Linear Spacesmentioning

confidence: 99%

Various Proofs of the Sylvester Criterion for Quadratic Forms

Giorgi

2017

JMR

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In the first part of the paper we present several proofs of the so-called Sylvester criterion for quadratic forms; some of the said proofs are short and easy. In the second part of the paper we give an algebraic proof of the Sylvester criterion for quadratic forms subject to a linear homogeneous system. KeywordsQuadratic forms, Sylvester criterion, positive definite quadratic forms, constrained quadratic forms.

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Positive Definite Matrices and Sylvester's Criterion

Cited by 112 publications

References 1 publication

Energy optimization in binary star systems: explanation for equal mass members in close orbits

Energy optimization in binary star systems: explanation for equal mass members in close orbits

Real wave propagation in the isotropic-relaxed micromorphic model

Various Proofs of the Sylvester Criterion for Quadratic Forms

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