M.Sc Thesis | |

M.Sc Student | Dina Michal |
---|---|

Subject | Spectrally Arbitrary Patterns |

Department | Department of Mathematics |

Supervisors | PROFESSOR EMERITUS Abraham Berman |

PROFESSOR EMERITUS Daniel Hershkowitz |

A *sign pattern matrix** *is
a matrix which has + ,- or 0 as its entries.

If *S* = (*s*_{ij})
is an *n×n* sign pattern matrix then the sign pattern class of *S*
is

Q(*S*) = { *A* = (*a*_{ij})
in *M*_{n}*(*R*)* : sgn(*a*_{ij}) = *s*_{ij}
for all i,j }, where *M*_{n}*(*R*)* is the set of all *n×n*
matrices over R and sgn(*a*_{ij}) is the sign of the number *a*_{ij}.

An *n×n* sign pattern matrix *S*,
where *n≥*2, is called *spectrally arbitrary pattern* *(SAP)*
if given any monic polynomial *p(x)* of degree *n* with real coefficients,
there is a matrix *A* in *Q(S)* such that the characteristic
polynomial of *A* is *p(x).*

A sign pattern *B *= (*b*_{ij})
is a *superpattern* of a sign pattern *A* = (*a*_{ij})
if *b*_{ij} = *a*_{ij} whenever *a*_{ij}≠0.

A sign pattern *A* is *minimally
spectrally arbitrary pattern (MSAP)* if it is spectrally arbitrary but is
not spectrally arbitrary if any nonzero entry of *A* is replaced by zero.

Two sign patterns *A* and *B*
are *equivalent* if *B* may be obtained from *A* by some
combination of negation, transposition, permutation similarity and signature
similarity. It is known that if *A* and *B* are two equivalent sign
patterns then *A* is a SAP iff *B* is a SAP.

A *combinatorially symmetric*
sign pattern matrix is a square sign pattern matrix *A* where *a*_{ij}
≠ 0 iff *a*_{ji} ≠ 0.

A *tree sign pattern (t.s.p.)*
matrix is a combinatorially symmetric sign pattern matrix whose undirected
graph is a tree (possibly with loops).

In the thesis we describe the Nilpotent-Jacobian Method which is used for proving that a certain sign pattern, as well as all of its superpatterns, are SAPs and give an overview of the sign patterns which are known to be SAPs/MSAPs.

We then refer to the tree sign
patterns. We give basic properties of the spectrally arbitrary tree sign
patterns and give an overview of the classification of the 2×2, 3×3* *and 4×4
spectrally arbitrary t.s.p..

**The new results in the thesis are:**

• New patterns are proven to be SAPs. In order to verify that those patterns are not equivalent to any of the known SAPs, I use a new result which enables us to verify that two sign patterns are not equivalent.

• A classification of all the 5×5 spectrally arbitrary tridiagonal t.s.p., excluding three patterns. The concept of Grobner Bases is used in this classification.