On the spread of certain normal matrices

Abstract

The spread of a matrix is defined as the maximum absolute value of the difference between any two eigenvalues of that matrix. Of interest, here is the spread of n × n normal matrices with entries restricted to a closed interval. In particular, we are interested in the classes of real symmetric matrices and real skew-symmetric matrices. For each of these classes, we investigate the maximum spread under certain conditions, such as small rank, and determine the structure of these matrices when their spread attains this maximum value.

Keywords:

• eigenvalue
• spread
• rank
• normal matrix
• symmetric matrix
• skew symmetric matrix

AMS Subject Classifications::

• 15A09
• 15A18
• 15A24

Acknowledgements

We wish to offer our sincere thanks to the referee for his/her careful reading of previous versions of this article and for the large number of helpful suggestions that led to many significant improvements of our work. Research supported in part by an NSERC research grant.