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Publication Date

Summer 2016

Degree Type

Thesis - Campus Access Only

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

Advisor

Wasin So

Subject Areas

Mathematics

Abstract

In linear algebra, one of the central results is the Jordan canonical form theorem which states that every complex $ n \times n $ matrix $ A $ is similar to an essentially unique Jordan matrix $ J_{A} $. In this thesis, we study the number of zero entries of matrices similar to a fixed $ n \times n $ matrix $ A $. For $ n=2, 3 $ we show that the Jordan matrix $ J_{A} $ has the largest number of zero entries among all matrices similar to $ A.$ For $ n=4, $ we show that the Jordan matrix $ J_{A} $ has the largest number of zero entries among all matrices similar to $ A $ if and only if $ A $ and $ A^T $ is not of the form

\begin{center}

$ P $

$\left[ \begin{array}{cccc}

0 & e & 0 & d \\

a & 0 & 0 & 0 \\

0 & b & 0 & 0 \\

0 & 0 & c & 0 \\

\end{array} \right] P^{T}$ \hspace{1cm} with $ ae^2 + 4bcd = 0 $

\end{center}

nor of the form

\begin{center}

$ P $

$\left[ \begin{array}{cccc}

0 & d & 0 & 0 \\

a & 0 & 0 & 0 \\

0 & c & 0 & b \\

0 & 0 & e & 0 \\

\end{array} \right] P^{T}$ \hspace{1cm} with $ ad = be $

\end{center}

for some permutation matrix $ P, $ and nonzero $ a,b,c,d,e. $ On the other hand, we report the result of R.A. Brualdi, P. Pei, and X. Zhan that $ J_{A} $ has the largest number of \textit{off-diagonal} zero entries among all the matrices similar to $ A. $

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