Diagonalizable matrix

Summary

In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that P^{-1}AP=D, or equivalently A = PDP^{-1}. (Such P, D are not unique.) For a finite-dimensional vector space V, a linear map T:V\to V is called diagonalizable if there exists an ordered basis of V consisting of eigenvectors of T. These definitions are equivalent: if T has a matrix representation T = PDP^{-1} as above, then the column vectors of P form a basis consisting of eigenvectors of T, and the diagonal entries of D are the corresponding eigenva

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On Sampling Lattices with Similarity Scaling Relationships

Thierry Blu, Dimitri Nestor Alice Van De Ville

We provide a method for constructing regular sampling lattices in arbitrary dimensions together with an integer dilation matrix. Subsampling using this dilation matrix leads to a similarity-transformed version of the lattice with a chosen density reduction. These lattices are interesting candidates for multidimensional wavelet constructions with a limited number of subbands.

LATP2009

Gyrokinetic calculations with the GS2 code of impurity transport in ASDEX Upgrade H-mode plasmas are presented. A method to separate the diagonal and off-diagonal terms in the quasi-linear flux computed by the gyrokinetic code for a trace impurity is introduced and applied. It is shown that in the experimental conditions of strong central electron cyclotron heating, unstable modes rotating in the electron diamagnetic direction are excited in the central region of the plasma. These are generated by the non-adiabatic response of passing electrons and feature extremely elongated eigenfunctions along the field line. The related fluctuations in the electrostatic potential generate an outward convection of the impurities. These theoretical findings are in agreement with the outward convection of laser ablated Si measured in these experimental conditions. In contrast, in the outer region of the plasma, as well as everywhere in the case of discharges without central electron heating, ion temperature gradient modes produce an impurity pinch directed inwards, which is also in agreement with the experimental observations.

2007

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The dilation matrix associated with the three-dimensional (3-D) face-centered cubic (FCC) sublattice is often considered to be the natural 3-D extension of the two-dimensional (2-D) quincunx dilation matrix. However, we demonstrate that both dilation matrices are of different nature: while the 2-D quincunx matrix is a similarity transform, the 3-D FCC matrix is not. More generally, we show that is impossible to obtain a dilation matrix that is a similarity transform and performs downsampling of the Cartesian lattice by a factor of two in more than two dimensions. Furthermore, we observe that the popular 3-D FCC subsampling scheme alternates between three different lattices: Cartesian, FCC, and quincunx. The latter one provides a less isotropic sampling density, a property that should be taken into account to properly orient 3-D data before processing using such a subsampling matrix.

IEEE2005