#### Date Thesis Awarded

5-2011

#### Access Type

Honors Thesis -- Access Restricted On-Campus Only

#### Degree Name

Bachelors of Science (BS)

#### Department

Mathematics

#### Advisor

Charles R. Johnson

#### Committee Members

John Delos

C. Ryan Vinroot

#### 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 $\{A_i\}_{i=1}^m$.

#### Recommended Citation

Yang, Dian, "Solution Theory for Systems of Bilinear Equations" (2011). *Undergraduate Honors Theses.* William & Mary. Paper 415.

https://scholarworks.wm.edu/honorstheses/415

#### Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License.

## Comments

Thesis is part of Honors ETD pilot project, 2008-2013. Migrated from Dspace in 2016.