Superpatterns of Spectrally Arbitrary Sign Patterns
Start Date
7-8-2026 11:00 AM
End Date
7-8-2026 11:15 AM
Location
ALT 205
Abstract
A sign pattern is an n x n matrix whose entries are in {+, -, 0}. A superpattern of a sign pattern is obtained by replacing some (or none) of its 0 entries by + or -. Any matrix whose real number entries follow that sign pattern is called a realization of the sign pattern. All matrices have a certain polynomial associated to them which is called a characteristic polynomial. For a given matrix, the roots of the characteristic polynomial are called eigenvalues and the collection of all eigenvalues is called the spectrum of the matrix. A spectrally arbitrary sign pattern is a sign pattern that has a matrix realization for any specified collection of eigenvalues.
In our research, we investigated methods to determine whether a sign pattern is spectrally arbitrary. The two common methods are the Nilpotent-Jacobian and Nilpotent-Centralizer methods. Both methods show that the sign pattern and all its superpatterns are spectrally arbitrary and rely on a special nilpotent realization which satisfies certain conditions.
In 2020, Michael Cavers, Jonathon Fischer, and Kevin N. Vander Meulen gave a family of spectrally arbitrary sign patterns that don’t have such special nilpotent matrix realizations. Their proof used a special polynomial technique in showing that all characteristic polynomials can be realized. However, this method did not prove that superpatterns of those patterns are spectrally arbitrary. Modifying their method, we demonstrated that special nilpotent realizations for some superpatterns exist and used the Nilpotent-Jacobian method to show they are spectrally arbitrary under certain conditions.
Superpatterns of Spectrally Arbitrary Sign Patterns
ALT 205
A sign pattern is an n x n matrix whose entries are in {+, -, 0}. A superpattern of a sign pattern is obtained by replacing some (or none) of its 0 entries by + or -. Any matrix whose real number entries follow that sign pattern is called a realization of the sign pattern. All matrices have a certain polynomial associated to them which is called a characteristic polynomial. For a given matrix, the roots of the characteristic polynomial are called eigenvalues and the collection of all eigenvalues is called the spectrum of the matrix. A spectrally arbitrary sign pattern is a sign pattern that has a matrix realization for any specified collection of eigenvalues.
In our research, we investigated methods to determine whether a sign pattern is spectrally arbitrary. The two common methods are the Nilpotent-Jacobian and Nilpotent-Centralizer methods. Both methods show that the sign pattern and all its superpatterns are spectrally arbitrary and rely on a special nilpotent realization which satisfies certain conditions.
In 2020, Michael Cavers, Jonathon Fischer, and Kevin N. Vander Meulen gave a family of spectrally arbitrary sign patterns that don’t have such special nilpotent matrix realizations. Their proof used a special polynomial technique in showing that all characteristic polynomials can be realized. However, this method did not prove that superpatterns of those patterns are spectrally arbitrary. Modifying their method, we demonstrated that special nilpotent realizations for some superpatterns exist and used the Nilpotent-Jacobian method to show they are spectrally arbitrary under certain conditions.
