The Mathematics of Chandler Davis

Holbrook, John

doi:10.1007/s00283-013-9432-2

Cited by 1 publication

(2 citation statements)

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“…Then the density for its real shadow has an expression in terms of a 2 F 1 -hypergeometric function which solves a certain second-order differential equation. Suppose a 1 = 0, so formulas ( 14) and (10)…”

Section: Shadows Of Hermitian and Real Symmetric Matricesmentioning

confidence: 99%

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Real numerical shadow and generalized B-splines

Dunkl

Gawron

Pawela

et al. 2015

Linear Algebra and its Applications

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Restricted numerical shadow $P^X_A(z)$ of an operator $A$ of order $N$ is a probability distribution supported on the numerical range $W_X(A)$ restricted to a certain subset $X$ of the set of all pure states - normalized, one-dimensional vectors in ${\mathbb C}^N$. Its value at point $z \in {\mathbb C}$ equals to the probability that the inner product $< u |A| u >$ is equal to $z$, where $u$ stands for a random complex vector from the set $X$ distributed according to the natural measure on this set, induced by the unitarily invariant Fubini-Study measure. For a Hermitian operator $A$ of order $N$ we derive an explicit formula for its shadow restricted to real states, $P^{\mathbb R}_A(x)$, show relation of this density to the Dirichlet distribution and demonstrate that it forms a generalization of the $B$-spline. Furthermore, for operators acting on a space with tensor product structure, ${\cal H}_A \otimes {\cal H}_B$, we analyze the shadow restricted to the set of maximally entangled states and derive distributions for operators of order N=4.Comment: 39 pages, 7 figure

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Section: Shadows Of Hermitian and Real Symmetric Matricesmentioning

confidence: 99%

“…Even though several papers on numerical shadow were published during the last five years [6,7,8], the idea to associate with the numerical range a probability measure is much older: as described in a recent review by Holbrook [10] it goes back to the early papers of Davis [1].…”

Section: Introductionmentioning

confidence: 99%