p . It follows that inequality (1.2) holds for any a ∈ lp when U1/p ≥ ||C||p,p and fails to hold for some a ∈ lp when U1/p < ||C||p,p. Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cn,k = 1/n, k ≤ n and 0 otherwise, is bounded on l p and has norm ≤ p/(p−1). (The norm is in fact p/(p− 1).) We say a matrix A = (an,k) is a lower triangular matrix if an,k = 0 for n < k...

In this paper, vectors are column vectors. Unless otherwise specified, the norm of a matrix ‖M‖ is the Frobenius norm, defined by ‖M‖ = √∑ i,jM 2 ij . The all ones vector is denoted 1 and the all zeros vector is denoted 0. The all ones matrix is denoted J and the all zeros matrix is denoted O. The n× n identity matrix is denoted In, or simply I if its size is understood from context. The vector...

p . It follows that inequality (1.2) holds for any a ∈ lp when U1/p ≥ ||C||p,p and fails to hold for some a ∈ lp when U1/p < ||C||p,p. Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cn,k = 1/n, k ≤ n and 0 otherwise, is bounded on l p and has norm ≤ p/(p−1). (The norm is in fact p/(p− 1).) We say a matrix A = (an,k) is a lower triangular matrix if an,k = 0 for n < k...

Introduction and notation. The sensitivity of computational problems is a major factor determining the accuracy of computations in machine arithmetic. It may be revealed and taken into account by the methods of perturbation analysis [14, 6]. Below we consider the technique of Lyapunov majorants for perturbation analysis of algebraic matrix equations F (A, X) = 0 arising in science and engineeri...

Let and be a sequence with non-negative entries. If , denote by the infimum of those satisfying the following inequality: whenever . The purpose of this paper is to give an upper bound for the norm of operator T on weighted sequence spaces d(w,p) and lp(w) and also e(w,?). We considered this problem for certain matrix operators such as Norlund, Weighted mean, Ceasaro and Copson ma...

Let A be a m × m complex matrix with zero trace. Then there are m ×m matrices B and C such that A = [B,C] and ‖B‖‖C‖2 ≤ (logm + O(1))‖A‖2 where ‖D‖ is the norm of D as an operator on `2 and ‖D‖2 is the Hilbert–Schmidt norm of D. Moreover, the matrix B can be taken to be normal. Conversely there is a zero trace m × m matrix A such that whenever A = [B,C], ‖B‖‖C‖2 ≥ | logm−O(1)|‖A‖2 for some abso...

In this paper we focus on the issue of normalization of the affinity matrix in spectral clustering. We show that the difference between N-cuts and Ratio-cuts is in the error measure being used (relative-entropy versus L1 norm) in finding the closest doubly-stochastic matrix to the input affinity matrix. We then develop a scheme for finding the optimal, under Frobenius norm, doubly-stochastic ap...

In this paper, we present a parallel quasi-Chebyshev acceleration applied to the nonoverlapping multisplitting iterative method for the linear systems when the coefficient matrix is either an H-matrix or a symmetric positive definite matrix. First, m parallel iterations are implemented in m different processors. Second, based on l1-norm or l2-norm, the m optimization models are parallelly treat...

We propose a convex optimization formulation with the Ky Fan 2-k-norm and `1-norm to find k largest approximately rank-one submatrix blocks of a given nonnegative matrix that has low-rank block diagonal structure with noise. We analyze low-rank and sparsity structures of the optimal solutions using properties of these two matrix norms. We show that, under certain hypotheses, with high probabili...

This paper is concerned with the determination of the Jordan canonical form and D1/,2-norm of the SOR iterative matrix derived from the coefficient matrix A having the form *-& t) with D\ and Di symmetric and positive definite. The theoretical results show that the Jordan form is not diagonal, but has only q principal vectors of grade 2 and that the D1//2-norm of J2?Ub (u^, the optimum paramete...