# Solution Theory for Systems of Bilinear Equations

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