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

Abstract

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.