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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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Bidiagonal Factorization

Bidiagonal Factorization

(p.43) Chapter Two Bidiagonal Factorization
Totally Nonnegative Matrices

Shaun M. Fallat

Charles R. Johnson

Princeton University Press

This chapter introduces and methodically develops the important and useful topic of bidiagonal factorization. Factorization of matrices is one of the most important topics in matrix theory, and plays a central role in many related applied areas such as numerical analysis and statistics. Investigating when a class of matrices admits a particular type of factorization is an important study, which historically has been fruitful. Often many intrinsic properties of a particular class of matrices can be deduced via certain factorization results. For example, it is a well-known fact that any (invertible) M-matrix can be factored into a product of a lower triangular (invertible) M-matrix and an upper triangular (invertible) M-matrix. This LU factorization result leads to the conclusion that the class of M-matrices is closed under Schur complementation, because of the connection between LU factorizations and Schur complements. This chapter focuses on triangular factorization extended beyond just LU factorization, however.

Keywords:   bidiagonal factorization, matrix theory, numerical analysis, LU factorization, triangular factorization, planar diagrams, statistics

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