Matrix Quasinorms Induced by Maximal and Minimal Vector Norms

Notes on Vector and Matrix Norms

Vector/Matrix Norms and Iterative Linear Solvers

a. ‖x‖ ≥ 0 for all x ∈ IRn, and ‖x‖ = 0 if and only if x is the zero vector. b. ‖αx‖ = |α|‖x‖. c. ‖x+ y‖ ≤ ‖x‖+ ‖y‖ [called the “triangle inequality”]. While this may seem a bit esoteric, consider that if x ∈ IR1, i.e. if x is a real number, the absolute value function, | · |, satisfies all of the above properties, and therefore constitutes as a norm. Another familiar norm would be the Euclidea...

MAXIMAL ALLOCATED BENEFIT AND MINIMAL ALLOCATED COST AND ITS APPLICATION

In this paper, we investigate the problems of consensus-making among institution in stock exchange with multiple criteria for evaluating performance when the players (institutions) are supposed to be egoistic and the score for each criterion for a player is supposed to be a positive score. Each player sticks to his superiority regarding the criteria. This paper introduces the models for computi...

On Sketching Matrix Norms and the Top Singular Vector

Sketching is a prominent algorithmic tool for processinglarge data. In this paper, we study the problem of sketchingmatrix norms. We consider two sketching models. The firstis bilinear sketching, in which there is a distribution overpairs of r×n matrices S and n× s matrices T such that forany fixed n×n matrix A, from S ·A ·T one can approximate‖A‖p up to an approxima...

Matrix Factorizations Based on Induced Norms

We decompose a matrix Y into a sum of rank one bilinear forms in a stepwise manner, by considering Y as a mapping from the finite dimensional space l r to the space l p . We provide transition formulas, and represent them in a duality diagram, thus generalizing the well known duality diagram in the french school of data analysis. As an application, we introduce a family of Euclidean multidimens...