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Solution Theory for Systems of Bilinear Equations

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Author: Yang, Dian
Advisor: Johnson, Charles R.
Committee Members: Delos, John; Vinroot, C. Ryan
Issued Date: 5/13/2011
Subjects: Bilinear system of equations
Bilinear forms
Rank one completion problem
Linear algebra
URI: http://hdl.handle.net/10288/13726
Description: 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 $\{A_i\}_{i=1}^m$.
Degree: Bachelors of Science in Mathematics

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