Abstract:

Are eigenvalues continuous functions of matrix entries? In other words, do the eigenvalues of a matrix change continuously as the matrix itself undergoes small changes? (The answer is yes or no, depending on what is meant by eigenvalue continuity).

Eigenvalue continuity is a fundamental concept in linear algebra and matrix theory that provides insights into the behavior of eigenvalues under small changes to matrices, with wide-ranging implications in theory and applications. However, eigenvalue continuity has been misused or carelessly used.

We discuss the continuity of eigenvalues as functions (functional continuity) as well as in the topological sense (topological continuity). This talk is also concerned with a few issues of linear algebra, both in teaching and research. These issues are usually overlooked or carelessly treated in texts or teaching.

Speaker:

Fuzhen Zhang, Ph.D., Professor, Nova Southeastern University, Fort Lauderdale, Florida, USA. zhang@nova.edu