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searching for Positive definiteness 53 found (60 total)

Pseudo-Riemannian manifold (1,171 words) [view diff] no match in snippet view article find links to article

generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold

Symmetric rank-one (1,149 words) [view diff] no match in snippet view article find links to article

x_{k}-H_{k}y_{k})^{T}y_{k}}}} . One might wonder why positive-definiteness is not preserved — after all, a rank-1 update of the form B k +

Absolute value (3,299 words) [view diff] exact match in snippet view article find links to article

generalization of this notion to other domains: Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition.

Positive harmonic function (861 words) [view diff] exact match in snippet view article find links to article

characterized in 1907 by Constantin Carathéodory in terms of the positive definiteness of their Taylor coefficients. A positive function f on the unit

Weil's criterion (254 words) [view diff] no match in snippet view article find links to article

the criterion can be used to show the corresponding statement of positive-definiteness does hold. A. Weil, "Sur les 'formules explicites' de la théorie

Davidon–Fletcher–Powell formula (991 words) [view diff] exact match in snippet view article find links to article

multidimensional problem. This update maintains the symmetry and positive definiteness of the Hessian matrix. Given a function f ( x ) {\displaystyle f(x)}

Asymmetric norm (705 words) [view diff] exact match in snippet view article find links to article

condition of positive definiteness is omitted, then p {\displaystyle p} is an asymmetric seminorm. A weaker condition than positive definiteness is non-degeneracy:

Normed vector space (2,890 words) [view diff] exact match in snippet view article find links to article

{\displaystyle x\in V} , ‖ x ‖ ≥ 0 {\displaystyle \;\lVert x\rVert \geq 0} . Positive definiteness: for every x ∈ V {\displaystyle x\in V} , ‖ x ‖ = 0 {\displaystyle

Schur complement (2,901 words) [view diff] no match in snippet view article find links to article

The Schur complement of a block matrix, encountered in linear algebra and the theory of matrices, is defined as follows. Suppose p, q are nonnegative integers

MINUIT (392 words) [view diff] no match in snippet view article find links to article

inexact line search, a stable metric updating scheme, and checks for positive-definiteness". The program is widely used in particle physics and thousands of

Finsler's lemma (1,587 words) [view diff] exact match in snippet view article find links to article

named after Paul Finsler. It states equivalent ways to express the positive definiteness of a quadratic form Q constrained by a linear form L. Since it is

INTLAB (834 words) [view diff] exact match in snippet view article find links to article

INTLAB can handle the least squares, Hessian matrix, and verify the positive definiteness of a given matrix) root-finding algorithm Affine arithmetic Solving

Period mapping (2,076 words) [view diff] exact match in snippet view article find links to article

FpHk(Xb, C) is orthogonal to Fk − p + 1Hk(Xb, C) with respect to Q. Positive definiteness: For all p + q = k, the restriction of ( − 1 ) k ( k − 1 ) / 2 i

Covariance function (582 words) [view diff] exact match in snippet view article find links to article

y ) = C s ( y − x ) . {\displaystyle C(x,y)=C_{s}(y-x).\,} The positive definiteness of this single-argument version of the covariance function can be

Inner product space (7,315 words) [view diff] no match in snippet view article find links to article

\langle ax+by,z\rangle =a\langle x,z\rangle +b\langle y,z\rangle .} Positive-definiteness: if x {\displaystyle x} is not zero, then ⟨ x , x ⟩ > 0 {\displaystyle

Statistical distance (643 words) [view diff] exact match in snippet view article find links to article

of indiscernibles. Note that condition 1 and 2 together produce positive definiteness) d(x, y) = d(y, x) (symmetry) d(x, z) ≤ d(x, y) + d(y, z)

Hopf maximum principle (495 words) [view diff] exact match in snippet view article find links to article

known to be continuous then it is sufficient to demand pointwise positive definiteness of aij on the domain. It is usually thought that the Hopf maximum

Hermitian manifold (1,507 words) [view diff] no match in snippet view article find links to article

is the complexification of a real form on TM. The symmetry and positive-definiteness of g on TM follow from the corresponding properties of h. In local

Finsler manifold (1,942 words) [view diff] exact match in snippet view article find links to article

necessarily for λ < 0) (positive homogeneity). F(v) > 0 unless v = 0 (positive definiteness). In other words, F(x, −) is an asymmetric norm on each tangent

P-adic valuation (1,366 words) [view diff] no match in snippet view article find links to article

Non-negativity | r | p ≥ 0 {\displaystyle |r|_{p}\geq 0} Positive-definiteness | r | p = 0 ⟺ r = 0 {\displaystyle |r|_{p}=0\iff r=0} Multiplicativity

Absolute value (algebra) (1,317 words) [view diff] exact match in snippet view article

{\displaystyle \left|x\right|=0} if and only if x = 0 {\displaystyle x=0} (positive definiteness) • | x y | = | x | | y | {\displaystyle \left|xy\right|=\left|x

Hermitian matrix (3,013 words) [view diff] exact match in snippet view article find links to article

they represent the relationships between different variables. The positive definiteness of a Hermitian covariance matrix ensures the well-definedness of

Bochner's theorem (1,406 words) [view diff] no match in snippet view article find links to article

{\displaystyle f} follows from the dominated convergence theorem. For positive-definiteness, take a nondegenerate representation of C 0 ( G ^ ) {\displaystyle

Gram matrix (2,683 words) [view diff] exact match in snippet view article find links to article

from the bi-linearity of the inner-product, and the last from the positive definiteness of the inner product. Note that this also shows that the Gramian

Minor (linear algebra) (2,799 words) [view diff] exact match in snippet view article

matrices, the leading principal minors can be used to test for positive definiteness and the principal minors can be used to test for positive semidefiniteness

Definite matrix (8,593 words) [view diff] exact match in snippet view article find links to article

known as Sylvester's criterion, and provides an efficient test of positive definiteness of a symmetric real matrix. Namely, the matrix is reduced to an

Quadratic programming (1,910 words) [view diff] no match in snippet view article find links to article

L ( x , λ ) = 0 {\displaystyle \nabla _{x}L(x,\lambda )=0} and positive-definiteness of Q: x ∗ = − Q − 1 A ⊤ λ . {\displaystyle x^{*}=-Q^{-1}A^{\top

SU(2) color superconductivity (921 words) [view diff] no match in snippet view article

potentials for even numbers of the quark flavors is associated with the positive-definiteness of the integral measure and the absence of a sign problem. QCD matter

Mercer's theorem (1,784 words) [view diff] no match in snippet view article find links to article

well-approximated by its Riemann sums, which are non-negative by positive-definiteness of K, implying λ ⟨ f , f ⟩ ≥ 0 {\displaystyle \lambda \langle f

Sylvester's criterion (1,591 words) [view diff] no match in snippet view article find links to article

Criterion for positive-definiteness of a matrix

GCD matrix (1,376 words) [view diff] exact match in snippet view article find links to article

1494695. S2CID 117112282. Haukkanen, P.; Toth, L. (2018). "Inertia, positive definiteness and ℓp norm of GCD and LCM matrices and their unitary analogs".

Covariance (4,738 words) [view diff] exact match in snippet view article find links to article

identification turns the positive semi-definiteness above into positive definiteness.) That quotient vector space is isomorphic to the subspace of random

Stationary process (2,627 words) [view diff] exact match in snippet view article find links to article

square-integrable random variables on the given probability space). By the positive definiteness of the autocovariance function, it follows from Bochner's theorem

Quasi-Newton method (2,264 words) [view diff] no match in snippet view article find links to article

The SR1 formula does not guarantee the update matrix to maintain positive-definiteness and can be used for indefinite problems. The Broyden's method does

Trace (linear algebra) (5,502 words) [view diff] no match in snippet view article

in the above formula, tr(A⊤B) = tr(B⊤A). These demonstrate the positive-definiteness and symmetry required of an inner product; it is common to call

Linear algebra (7,979 words) [view diff] no match in snippet view article find links to article

\rangle +\langle \mathbf {v} ,\mathbf {w} \rangle .\end{aligned}}} Positive-definiteness: ⟨ v , v ⟩ ≥ 0 {\displaystyle \langle \mathbf {v} ,\mathbf {v} \rangle

Contraction (operator theory) (2,901 words) [view diff] no match in snippet view article

{\Phi (g)=PU(g)P,}} with values in operators on K satisfies the positive-definiteness condition ∑ λ i λ j ¯ Φ ( g j − 1 g i ) = P T ∗ T P ≥ 0 , {\displaystyle

Zonal spherical function (6,693 words) [view diff] exact match in snippet view article find links to article

class of zonal spherical functions can be obtained by dropping positive definiteness from the conditions, but for these functions there is no longer

Modern portfolio theory (7,909 words) [view diff] exact match in snippet view article find links to article

with potential caveats (poor numerical accuracy, requirement of positive definiteness of the covariance matrix...). An alternative approach to specifying

Low-rank approximation (3,855 words) [view diff] exact match in snippet view article find links to article

categorical. Distance matrix completion, in which case there is a positive definiteness constraint. Natural language processing, in which case the approximation

Positive-definite kernel (4,244 words) [view diff] exact match in snippet view article find links to article

) ) F . {\displaystyle K(x,y)=(\Phi (x),\Phi (y))_{F}.} Indeed, positive definiteness of K {\displaystyle K} follows from the p.d. property of the inner

Multivariate normal distribution (9,519 words) [view diff] no match in snippet view article find links to article

&b_{n}\end{bmatrix}}=\mathbf {b} ^{\rm {T}}.} Observe how the positive-definiteness of Σ implies that the variance of the dot product must be positive

Mixture model (7,773 words) [view diff] exact match in snippet view article find links to article

the probability vector, and for sufficiently large sample sizes positive definiteness of the covariance iterates. This is a key advantage since explicitly

Mixture model (7,773 words) [view diff] exact match in snippet view article find links to article

the probability vector, and for sufficiently large sample sizes positive definiteness of the covariance iterates. This is a key advantage since explicitly

Conjugate gradient method (7,322 words) [view diff] exact match in snippet view article find links to article

decomposition of the preconditioner must be used to keep the symmetry (and positive definiteness) of the system. However, this decomposition does not need to be

Minkowski space (10,620 words) [view diff] exact match in snippet view article find links to article

modification (Lee deals with Riemannian metrics.). Where Lee refers to positive definiteness to show the injectivity of the map, one needs instead appeal to

Locally convex topological vector space (10,568 words) [view diff] exact match in snippet view article find links to article

{\displaystyle p(x+y)\leq p(x)+p(y).} If p {\displaystyle p} satisfies positive definiteness, which states that if p ( x ) = 0 {\displaystyle p(x)=0} then x

Reproducing kernel Hilbert space (6,334 words) [view diff] exact match in snippet view article find links to article

defines a kernel via Clearly K {\displaystyle K} is symmetric and positive definiteness follows from the properties of inner product in F {\displaystyle

Alain Goriely (2,747 words) [view diff] exact match in snippet view article find links to article

development of new geometric methods to prove stability through the positive definiteness of the second variation. With colleagues, he provided a complete

Orthotropic material (5,014 words) [view diff] exact match in snippet view article find links to article

J. Mech. Appl. Math., 58(1), pp. 73-82. Ting, T. C. T. (1996), "Positive definiteness of anisotropic elastic constants", Mathematics & Mechanics of Solids

Neural tangent kernel (5,156 words) [view diff] no match in snippet view article find links to article

converges to that minimum as t → ∞ {\displaystyle t\to \infty } . This positive-definiteness property has been shown in a number of cases, yielding the first

Glossary of real and complex analysis (3,630 words) [view diff] exact match in snippet view article find links to article

{\displaystyle x\in V} , ‖ x ‖ ≥ 0 {\displaystyle \;\lVert x\rVert \geq 0} . Positive definiteness: for every x ∈ V {\displaystyle x\in V} , ‖ x ‖ = 0 {\displaystyle

Symmetric cone (16,608 words) [view diff] no match in snippet view article find links to article

product for which the L(a) are self-adjoint than verify directly positive-definiteness of the trace form. (The equivalent original condition of Jordan