*Shaun M. Fallat and Charles R. Johnson*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691121574
- eISBN:
- 9781400839018
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691121574.003.0001
- Subject:
- Mathematics, Applied Mathematics

This introductory chapter is an overview of totally positive (or nonnegative) matrices (TP or TN matrices). Positivity has roots in every aspect of pure, applied, and numerical mathematics. The ...
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This introductory chapter is an overview of totally positive (or nonnegative) matrices (TP or TN matrices). Positivity has roots in every aspect of pure, applied, and numerical mathematics. The subdiscipline, total positivity, also is entrenched in nearly all facets of mathematics. At first it may appear that the notion of total positivity is artificial; however, this class of matrices arises in a variety of important applications. Historically, the theory of totally positive matrices originated from the pioneering work of Gantmacher and Krein in 1960. The chapter explores the extant literature on total positivity since then, before proceeding to the definitions and notations to be used in the rest of this volume. It also provides a brief overview of the succeeding chapters.Less

This introductory chapter is an overview of totally positive (or nonnegative) matrices (TP or TN matrices). Positivity has roots in every aspect of pure, applied, and numerical mathematics. The subdiscipline, total positivity, also is entrenched in nearly all facets of mathematics. At first it may appear that the notion of total positivity is artificial; however, this class of matrices arises in a variety of important applications. Historically, the theory of totally positive matrices originated from the pioneering work of Gantmacher and Krein in 1960. The chapter explores the extant literature on total positivity since then, before proceeding to the definitions and notations to be used in the rest of this volume. It also provides a brief overview of the succeeding chapters.

*Shaun M. Fallat and Charles R. Johnson*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691121574
- eISBN:
- 9781400839018
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691121574.003.0011
- Subject:
- Mathematics, Applied Mathematics

This chapter reviews a number of subtopics connected with TN matrices; including powers and roots of TN matrices, TP/TN polynomial matrices, subdirect sums of TN matrices, and Perron complements of ...
More

This chapter reviews a number of subtopics connected with TN matrices; including powers and roots of TN matrices, TP/TN polynomial matrices, subdirect sums of TN matrices, and Perron complements of TN matrices. Given that total positivity as well as TP matrices arise in many parts of mathematics and have a broad history, there have been many facts and properties about TP and TN matrices that neither fit well in another chapter nor are lengthy enough to constitute chapters in their own right. Yet, these topics are worthy of being mentioned and discussed. Hence the need for this final chapter, which represents a catalog of additional topics dealing with other aspects of TN matrices.Less

This chapter reviews a number of subtopics connected with TN matrices; including powers and roots of TN matrices, TP/TN polynomial matrices, subdirect sums of TN matrices, and Perron complements of TN matrices. Given that total positivity as well as TP matrices arise in many parts of mathematics and have a broad history, there have been many facts and properties about TP and TN matrices that neither fit well in another chapter nor are lengthy enough to constitute chapters in their own right. Yet, these topics are worthy of being mentioned and discussed. Hence the need for this final chapter, which represents a catalog of additional topics dealing with other aspects of TN matrices.

*Shaun M. Fallat and Charles R. Johnson*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691121574
- eISBN:
- 9781400839018
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691121574.003.0005
- Subject:
- Mathematics, Applied Mathematics

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 ...
More

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.Less

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.