The Equivalence of Definitions of a Matric Function

Rinehart, R. F.

doi:10.2307/2306996

Cited by 38 publications

(5 citation statements)

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“…There are several di¬erent ways to de ne an isotropic tensor function (see, for example, Rinehart (1955) for the comparison of various de nitions of matrix functions). An isotropic symmetric tensor function G(A) : Sym n !…”

Section: G(a);mentioning

confidence: 99%

Application of the Dunford-Taylor integral to isotropic tensor functions and their derivatives

Itskov

2003

Proc. R. Soc. Lond. A

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In this paper we obtain closed-form representations for isotropic tensor functions and their derivative using the Dunford{Taylor integral. These representations are given in terms of eigenvalues of the tensor argument and are valid for all second-order tensors de ned as a linear mapping over the n-dimensional real vector space. Our solution is especially useful for non-symmetric tensor functions, where representations based on the spectral decomposition in diagonal form cannot generally be applied. The de nition of an isotropic tensor function by the Dunford{Taylor integral appears to be advantageous over the tensor power series.

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Section: G(a);mentioning

confidence: 99%

Application of the Dunford-Taylor integral to isotropic tensor functions and their derivatives

Itskov

2003

Proc. R. Soc. Lond. A

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“…There exist several different definitions for matrix functions. The relationshop between these definitions is discussed in detail in [5]. We have chosen the most general definition that allows to express the function values by polynomials.…”

Section: Preliminariesmentioning

confidence: 99%

Engineering functional quantum algorithms

Klappenecker

Rötteler

2003

Phys. Rev. A

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Suppose that a quantum circuit with K elementary gates is known for a unitary matrix U , and assume that U m is a scalar matrix for some positive integer m. We show that a function of U can be realized on a quantum computer with at most O(mK + m 2 log m) elementary gates. The functions of U are realized by a generic quantum circuit, which has a particularly simple structure. Among other results, we obtain efficient circuits for the fractional Fourier transform.

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“…The definitions of differentiability and derivative shall be applicable and meaningful when applied to the special functions on Mq arising from scalar functions of a complex variable [2]. For example, it would be desirable that the function ez turn out to be differentiable, according to the general definition of differentiability of functions on Mq.…”

Section: IImentioning

confidence: 99%

“…on Mc [2 ]. Such a function/(Z) is defined at a matrix A, if f(z) is defined at the characteristic roots of A, and is analytic at those characteristic roots of multiplicity greater than one in the minimum equation of A.…”

Section: And MLmentioning

confidence: 99%

“…Since the characteristic roots of a matrix are continuous functions of the elements of the matrix, there exists a S>0 such that for all H with Norm H<8, the characteristic roots of A+H lie within <r. Let H be henceforth so restricted. Since/(z) is analytic in and on a, the Cauchy integral formula ior f(A+H) and f(A) is valid [2]; hence, letting a[ denote the boundary of <r,-, The elements of (zI -A -77)_1 and (zi-A)'1 are rational functions of z whose denominators are, or can be taken to be, det (zI-A -77)…”

Section: And MLmentioning

confidence: 99%

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