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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. $

#### Recommended Citation

Vu, MinhNhat, "A Study on the Sparsity Property of the Jordan Canonical Form" (2016). *Master's Theses*. 4741.

https://scholarworks.sjsu.edu/etd_theses/4741