A Linear Algebra Story: How We Reconstructed a Matrix from its Eigenvalues

Summary

By telling this story, we discuss how to blend research problems into the classrooms to enhance the curriculum starting already with the linear algebra course. In particular, we present an elementary approach to the reconstruction of persymmetric Jacobi matrices from their eigenvalues. This work is done in collaboration with and for undergraduate students taking linear algebra. We wrote it with a thought in mind to fill the gaps and to show the depth of results still keeping it within the boundaries of the undergraduate underclassman level. So if a student reads it and the web supplements, it will be completely understandable and by comparing to the original paper [2] the student would realize how to break down the results and proofs to fully comprehend them. As for educators, we hope this would show an example of how to present an establish knowledge and modern theories by making a problem feasible for the broader undergraduate audience. We also briefly discuss the motivation of studying the problem in the prism of modern theories. Namely, we link this algorithm to the perfect quantum transfer problem.

Acknowledgment

The second author wishes to thank their linear algebra students who kept working on their projects in the midst of the COVID-19 pandemic and survived the switch to the distance learning. They were an incredible source of inspiration.

Additional information

Notes on contributors

Alexander Holley

Alexander Holley ([email protected]) is pursuing a B.S. in Mechanical Engineering at the University of Connecticut. He plans on working in the aerospace field after graduation, or returning to school for a M. Sc. in a related engineering field. His interests include being in nature, cooking, and watching or playing sports.

Anastasiia Minenkova

Anastasiia Minenkova ([email protected]) is a visiting lecturer at Mount Holyoke College. She received her M.Sc. in Mathematics from the University of Mississippi and her doctorate from the University of Connecticut. Her research interests are in numerical linear algebra. She is very passionate about teaching. Anastasiia’s interests outside of academia include cross-stitching and culinary tourism.