## Master's Theses

Summer 2016

#### Degree Type

Thesis - Campus Access Only

#### Degree Name

Master of Science (MS)

#### Department

Mathematics and Statistics

Wasin So

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 $A.$

COinS

#### DOI

https://doi.org/10.31979/etd.7em8-5jdu