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Totally Nonnegative Matrices$
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Shaun M. Fallat and Charles R. Johnson

Print publication date: 2011

Print ISBN-13: 9780691121574

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691121574.001.0001

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The Spectral Structure of TN Matrices

The Spectral Structure of TN Matrices

Chapter:
(p.97) Chapter Five The Spectral Structure of TN Matrices
Source:
Totally Nonnegative Matrices
Author(s):

Shaun M. Fallat

Charles R. Johnson

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691121574.003.0006

This chapter highlights the spectral structure of TN matrices. By “spectral structure” this chapter refers to facts about the eigenvalues and eigenvectors of matrices in a particular class. In view of the well-known Perron-Frobenius theory that describes the spectral structure of general entrywise nonnegative matrices, it is not surprising that TN matrices have an important and interesting special spectral structure. Nonetheless, that spectral structure is remarkably strong, identifying TN matrices as a very special class among nonnegative matrices. All eigenvalues are nonnegative real numbers, and the sign patterns of the entries of the eigenvectors are quite special as well. This spectral structure is most apparent in the case of IITN matrices (that is, the classical oscillatory case originally described by Gantmacher and Krein).

Keywords:   TN matrices, eigenvalues, eigenvectors, Perron-Frobenius theory, IITN matrices

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