| Summary: | 碩士 === 國立中山大學 === 應用數學研究所 === 84 === In this thesis, we will study nilpotent matrices. A matrix $A$
is nilpotent if $A^r=0$ for some positive integer $r,$ and the
smallest $r$ is called the index of $A.$ First, we survey some
known results about nilpotent matrices. Then, we discuss the
index of certain real nilpotent matrices and give the best
bounds of their index. A sign pattern matrix requires (allows)
nilpotent if every (some) realmatrix in the associated class is
nilpotent. Similar to real nilpotent matrix of rank $k,$ we
give the necessary and sufficient condition for ''\emph{type k''
} sign pattern matrices that require (allow) nilpotent. In the
end, we give a method to construct ''\emph{type I''} and
''\emph{type II''}requiring (allowing) nilpotent sign pattern
matrices.
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