It is common to assume that a Hookean solid is isotropic. For a generally anisotropic elasticity tensor, it is possible to obtain its isotropic counterparts. Such a counterpart is obtained in accordance with a given norm. Herein, we examine the effect of three norms: the Frobenius 36-component norm, the Frobenius 21-component norm and the operator norm on a general Hookean solid. We find that b...

In the present paper‎, ‎we propose an iterative algorithm for‎ ‎solving the generalized $(P,Q)$-reflexive solution of the quaternion matrix‎ ‎equation $overset{u}{underset{l=1}{sum}}A_{l}XB_{l}+overset{v} ‎{underset{s=1}{sum}}C_{s}widetilde{X}D_{s}=F$‎. ‎By this iterative algorithm‎, ‎the solvability of the problem can be determined automatically‎. ‎When the‎ ‎matrix equation is consistent over...

We show that given an estimate  that is close to a general high-rank positive semidefinite (PSD) matrix A in spectral norm (i.e., ‖Â−A‖2 ≤ δ), the simple truncated Singular Value Decomposition of  produces a multiplicative approximation of A in Frobenius norm. This observation leads to many interesting results on general high-rank matrix estimation problems: 1. High-rank matrix completion: we...

Perturbation bounds for eigenvalues of diagonalizable matrices are derived that do not depend on any quantities associated with the perturbed matrix; in particular the perturbed matrix can be defective. Furthermore, Gerschgorin-like inclusion regions in the Frobenius are derived, as well as bounds on the departure from normality. 1. Introduction. The results in this paper are based on two eigen...

Nomenclature Hadamard product k k2, k kF spectral norm, Frobenius norm 0l m; 0l l m zero matrix, 0l l Il; 1l m l l identity matrix, l m ones matrix

This article introduces a systematic study for computational aspects of classical wavelet transforms over finite fields using tools from computational harmonic analysis and also theoretical linear algebra. We present a concrete formulation for the Frobenius norm of the classical wavelet transforms over finite fields. It is shown that each vector defined over a finite field can be represented as...

Given a primitive integer vector a ∈ Z>0, the largest integer b such that the knapsack polytope P = {x ∈ R≥0 : 〈a,x〉 = b} contains no integer point is called the Frobenius number of a. We show that the asymptotic growth of the Frobenius number in average is significantly slower than the growth of the maximum Frobenius number. More precisely, we prove that it does not essentially exceed ||a|| 1+...

An algorithm for matrix approximation, when only some of its entries are taken into consideration, is described. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank approximations, this type of algorithms appears recently in the literature under different names, where it usually uses the Expectation-Maximization algorithm that maximizes...