79 research outputs found
Generalized Hadamard Product and the Derivatives of Spectral Functions
In this work we propose a generalization of the Hadamard product between two
matrices to a tensor-valued, multi-linear product between k matrices for any . A multi-linear dual operator to the generalized Hadamard product is
presented. It is a natural generalization of the Diag x operator, that maps a
vector into the diagonal matrix with x on its main diagonal.
Defining an action of the orthogonal matrices on the space of
k-dimensional tensors, we investigate its interactions with the generalized
Hadamard product and its dual. The research is motivated, as illustrated
throughout the paper, by the apparent suitability of this language to describe
the higher-order derivatives of spectral functions and the tools needed to
compute them. For more on the later we refer the reader to [14] and [15], where
we use the language and properties developed here to study the higher-order
derivatives of spectral functions.Comment: 24 page
The higher-order derivatives of spectral functions
AbstractWe are interested in higher-order derivatives of functions of the eigenvalues of real symmetric matrices with respect to the matrix argument. We describe a formula for the k-th derivative of such functions in two general cases.The first case concerns the derivatives of the composition of an arbitrary (not necessarily symmetric) k-times differentiable function with the eigenvalues of symmetric matrices at a symmetric matrix with distinct eigenvalues.The second case describes the derivatives of the composition of a k-times differentiable separable symmetric function with the eigenvalues of symmetric matrices at an arbitrary symmetric matrix. We show that the formula significantly simplifies when the separable symmetric function is k-times continuously differentiable.As an application of the developed techniques, we re-derive the formula for the Hessian of a general spectral function at an arbitrary symmetric matrix. The new tools lead to a shorter, cleaner derivation than the original one.To make the exposition as self contained as possible, we have included the necessary background results and definitions. Proofs of the intermediate technical results are collected in the appendices
Higher derivatives of spectral functions associated with one-dimensional schrodinger operators
We investigate the existence and asymptotic behaviour of higher derivatives of the spectral function in the context of one-dimensional Schr¨odinger operators on the half-line with integrable potentials. In particular, we identify sufficient conditions on the potential for the existence and continuity of the n-th derivative, and outline a systematic procedure for estimating numerical upper bounds for the turning points of such derivatives. Explicit worked examples illustrate the development and application of the theory
Self-Consistent Perturbation Theory for Thermodynamics of Magnetic Impurity Systems
Integral equations for thermodynamic quantities are derived in the framework
of the non-crossing approximation (NCA). Entropy and specific heat of 4f
contribution are calculated without numerical differentiations of thermodynamic
potential. The formulation is applied to systems such as PrFe4P12 with
singlet-triplet crystalline electric field (CEF) levels.Comment: 3 pages, 2 figures, proc. ASR-WYP-2005 (JAERI
Locally symmetric submanifolds lift to spectral manifolds
In this work we prove that every locally symmetric smooth submanifold gives
rise to a naturally defined smooth submanifold of the space of symmetric
matrices, called spectral manifold, consisting of all matrices whose ordered
vector of eigenvalues belongs to the locally symmetric manifold. We also
present an explicit formula for the dimension of the spectral manifold in terms
of the dimension and the intrinsic properties of the locally symmetric
manifold
Isotropic functions revisited
To a smooth and symmetric function defined on a symmetric open set
and a real -dimensional vector space we
assign an associated operator function defined on an open subset
of linear transformations of , such that for
each inner product on , on the subspace
of -selfadjoint operators,
is the isotropic function associated to , which
means that , where denotes the
ordered -tuple of real eigenvalues of . We extend some well known
relations between the derivatives of and each to relations between
and . By means of an example we show that well known regularity
properties of do not carry over to .Comment: 13 pages. Added an example to show that loss of regularity is
possible. Extended the bibliography. Comments are welcom
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