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Solution Theory for Systems of Bilinear Equations
Yang, Dian
Yang, Dian
Abstract
For $A_1,\ldots , A_m\in M_{p,q}(\mathbb{F})$ and $g\in\mathbb{F}^m$, any system of equations of the form $y^TA_ix=g_i$, $i=1,\ldots, m$, with $y$ varying over $\mathbb{F}^p$ and $x$ varying over $\mathbb{F}^q$ is called bilinear. A solution theory for complete systems ($m=pq$) is given in \cite{MR2567143}. In this paper we give a general solution theory for bilinear systems of equations. For this, we notice a relationship between bilinear systems and linear systems. In particular we prove that the problem of solving a bilinear system is equivalent to finding rank one points of an affine matrix function. And we study how in general the rank one completion problem can be solved. We also study systems with certain left hand side matrices $\{A_i\}_{i=1}^m$ such that a solution exist no matter what right hand side $g$ is. Criteria are given to distinguish such $\{A_i\}_{i=1}^m$.
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Thesis is part of Honors ETD pilot project, 2008-2013. Migrated from Dspace in 2016.
Date
2011-05-13
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Keywords
Bilinear system of equations, Bilinear forms, Rank one completion problem, Linear algebra
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Department
Mathematics