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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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Sign Variation of Vectors and TN Linear Transformations

Sign Variation of Vectors and TN Linear Transformations

Chapter:
(p.87) Chapter Four Sign Variation of Vectors and TN Linear Transformations
Source:
Totally Nonnegative Matrices
Author(s):

Shaun M. Fallat

Charles R. Johnson

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691121574.003.0005

This chapter develops an array of results about TP/TN matrices and sign variation diminution, along with appropriate converses. If the entries of a vector represent a sampling of consecutive function values, then a change in the sign of consecutive vector entries corresponds to the important event of the (continuous) function passing through zero. It has been known that as a linear transformation, a TP matrix cannot increase the number of sign changes in a vector. The transformations that never increase the number of sign variations are of interest in a variety of applications, including approximation theory and shape preserving transforms, and are of mathematical interest as well.

Keywords:   TN linear transformations, sign variation diminution, vectors, linear transformations, TP matrix, EB factorization

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