منابع مشابه
Displacement Ranks of a Matrix
The solution of many problems in physics and engineering reduces ultimately to the solution of linear equations of the form Ra = m, where JR and m are given N x N and N x 1 matrices and a is to be determined. Here our concern is with the fact that it generally takes 0(N) computations (one computation being the multiplication of two real numbers) to do this, and this might be a substantial burde...
متن کاملOn the complexity of Boolean matrix ranks
We construct a reduction which proves that the fooling set number and the determinantal rank of a Boolean matrix are NP-hard to compute. This note is devoted to the functions of determinantal rank and fooling set number, which are receiving attention in different applications, see [1, 3] and references therein. The purpose of this note is to give an NP-completeness proof for those functions, th...
متن کاملRanks of the common solution to some quaternion matrix equations with applications
We derive the formulas of the maximal andminimal ranks of four real matrices $X_{1},X_{2},X_{3}$ and $X_{4}$in common solution $X=X_{1}+X_{2}i+X_{3}j+X_{4}k$ to quaternionmatrix equations $A_{1}X=C_{1},XB_{2}=C_{2},A_{3}XB_{3}=C_{3}$. Asapplications, we establish necessary and sufficient conditions forthe existence of the common real and complex solutions to the matrixequations. We give the exp...
متن کاملMatrix Algebras and Displacement Decompositions
A class ξ of algebras of symmetric n × n matrices, related to Toeplitz-plus-Hankel structures and including the well-known algebra H diagonalized by the Hartley transform, is investigated. The algebras of ξ are then exploited in a general displacement decomposition of an arbitrary n× n matrix A. Any algebra of ξ is a 1-space, i.e., it is spanned by n matrices having as first rows the vectors of...
متن کاملFast Estimation of Approximate Matrix Ranks Using Spectral Densities
Many machine learning and data-related applications require the knowledge of approximate ranks of large data matrices at hand. This letter presents two computationally inexpensive techniques to estimate the approximate ranks of such matrices. These techniques exploit approximate spectral densities, popular in physics, which are probability density distributions that measure the likelihood of fi...
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ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 1979
ISSN: 0273-0979
DOI: 10.1090/s0273-0979-1979-14659-7