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Extrapolated positive definite and positive semi-definite splitting methods for solving non-Hermitian positive definite linear systems

Applications of Mathematics, 2021
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Shokrpour, Raheleh, Ebadi, Ghodrat
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On Different Definitions of Positive Definiteness

Mathematische Nachrichten, 1991
Let the function \(T^*\) be defined by \(T^*(a)=T(a^{-1})^*\) for a function \(T\) of a group \(G\) into a Hilbert space. The authors show that the commutativity of \(G\) is equivalent to each of the conditions below: (1) \(T^*\) is positive definite for every representation \(T\) of \(G\) in Hilbert space.
Friedrich, J., Klotz, L.
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Positive Definite Toeplitz Completions

Journal of the London Mathematical Society, 1999
Consider a partially prescribed Toeplitz matrix \(T\), that is, a Toeplitz matrix in which some diagonals are specified, while other are unspecified and may thus be treated as free variables. A completion of \(T\) is an assignment of values to the unspecified diagonals. The pattern \(P(T)\) of a partially defined Hermitian Toeplitz matrix is the set of
Johnson, Charles R., Lundquist, Michael, Nævdal, Geir   +2 more
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Positive Powers of Positive Positive Definite Matrices

Canadian Journal of Mathematics, 1996
AbstractLet C be an n x n positive definite matrix. If C ≥ 0 in the sense that Cij ≥ 0 and if p > n — 2, then Cp ≥ 0. This implies the following "positive minorant property" for the norms ‖A‖p = [tr(A*A)p/2]1/P. Let 2 < p ≠ 4, 6, … . Then 0 ≤ A ≤ B => ‖A‖p ≥ ‖B‖P if and only if n < p/2 + 1.
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Unbounded Positive Definite Functions

Canadian Journal of Mathematics, 1969
Let G be an abelian group, written additively. A complexvalued function ƒ, defined on G, is said to be positive definite if the inequality1holds for every choice of complex numbers C1, …, cn and S1, …, sn in G. It follows directly from (1) that every positive definite function is bounded. Weil (9, p.
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Verification of Positive Definiteness

BIT Numerical Mathematics, 2006
An efficient numerical criterion to verify positive definiteness of a symmetric or Hermitian matrix is presented. The criterion is based on standard IEEE 754 floating-point arithmetic with rounding to nearest. It implements a single floating-point Cholesky decomposition and improves a known result.
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On the Measurability of Positive Definite and Conditionally Positive Definite Functions

Mathematische Nachrichten, 1986
Let f be a positive definite function on a locally compact abelian group G. In this paper we show that measurability of f on an open neighbourhood of the zero implies measurability of f on G. The same result holds for conditionally positive definite functions.
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Positive Definite Matrices

The American Mathematical Monthly, 1970
(1970). Positive Definite Matrices. The American Mathematical Monthly: Vol. 77, No. 3, pp. 259-264.
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Complex Variable Positive Definite Functions

Complex Analysis and Operator Theory, 2013
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Buescu, Jorge, Paixão, António
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Positive Definite $C^p $ Kernels

SIAM Journal on Mathematical Analysis, 1986
Let K(x,t) be a positive definite Hermitian \(C^ p\) kernel on \(| x| \leq 1\), \(| t| \leq 1\); and \(e_ 1\geq e_ 2\geq...\geq e_ n\geq..\). the associated eigenvalues. \textit{Chungwei Ha} [ibid.
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