Totally Nonnegative Matrices

Shaun M. Fallat
Charles R. Johnson
Pages: 264
https://www.jstor.org/stable/j.ctt7scff

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. List of Figures
(pp. xi-xii)
4. Preface
(pp. xiii-xvi)
SMF and CRJ
5. Chapter Introduction
(pp. 1-26)

The central theme in this book is to investigate and explore various properties of the class oftotally nonnegative matrices.

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. For example, totally positive (or nonnegative) matrices arise in statistics (see [GM96, Goo86, Hei94]), mathematical biology (see [GM96]), combinatorics (see [BFZ96, Bre95, Gar96, Peñ98]), dynamics (see [GK02]), approximation theory (see [GM96, Pri68, CP94b]), operator theory (see [Sob75]), and geometry (see [Stu88]). Historically, the theory of totally positive matrices originated from the pioneering work of Gantmacher...

6. Chapter One Preliminary Results and Discussion
(pp. 27-42)

Along with the elementary bidiagonal factorization, to be developed in the next chapter, rules for manipulating determinants and special determinantal identities constitute the most useful tools for understanding TN matrices. Some of this technology is simply from elementary linear algebra, but the less well-known identities are given here for reference. In addition, other useful background facts are entered into the record, and a few elementary and frequently used facts about TN matrices are presented. Most are stated without proof, but are accompanied by numerous references where proofs and so on, may be found. The second component features more modern results,...

7. Chapter Two Bidiagonal Factorization
(pp. 43-72)

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. In fact, a typical (and useful) preliminary topic for any course in linear algebra is Gaussian elimination and elementary row operations, which essentially represents the groundwork for many forms of matrix factorization. In particular, triangular factorizations are a byproduct of many sorts of elimination strategies including Gaussian elimination.

Investigating when a class of matrices (such as positive definite or TP) admit a particular type of factorization is an important study, which...

8. Chapter Three Recognition
(pp. 73-86)

When TN matrices are first encountered, it may seem that examples should be rate and that checking for TN or TP is tedious and time consuming.

However, TN matrices enjoy tremendous structure, as a result of requiring all minors to be nonnegative. This intricate structure actually makes it easier to determine when a matrix is TP than to check when it is aP-matrix, which formally involves far fewer minors.

Furthermore, this structure is one of the reasons that TN matrices arise in many applications in both pure and applied mathematics. We touch on one such application here, which also...

9. Chapter Four Sign Variation of Vectors and TN Linear Transformations
(pp. 87-96)

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. Since [Sch30], it has been known that as a linear transformation, a TP matrix cannot increase the number of sign changes in a vector (this has a clear meaning when bothxandAxare totally nonzero and will be given an appropriate general meaning later). The transformations that never increase the number of sign variations are of interest in a variety of applications, including...

10. Chapter Five The Spectral Structure of TN Matrices
(pp. 97-128)

By “spectral structure” we mean 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...

11. Chapter Six Determinantal Inequalities for TN Matrices
(pp. 129-152)

In this chapter we begin an investigation into the possible relationships among the principal minors of TN matrices. A natural and important goal is to examine all the inequalities among the principal minors of TN matrices. More generally, we also study families of determinantal inequalities that hold for all TN matrices.

In 1893, Hadamard noted the fundamental determinantal inequality$$\det A \leq \prod_{i=1}^n a_{ii} \hspace {70pt} (6.1)$$holds wheneverA= [aij] is a positive semidefinite Hermitian matrix. Hadamard observed that it is not difficult to determine that (6.1) is equivalent to the inequality$$|\det A| \leq \sqrt {\prod_{i=1}^n \left( \sum_{j = 1}^n | a_{ij} |^2 \right)}, \hspace{70pt}(6.2)$$whereA= [aij] is an arbitraryn-by-nmatrix.

12. Chapter Seven Row and Column Inclusion and the Distribution of Rank
(pp. 153-166)

Recall that anm-by-nmatrixAis said to berank deficientif rankA< min{m,n}.

Just as with zero entries (see Section 1.6), the distribution of ranks among submatrices of a TN matrix is much less free than in a general matrix. Rank deficiency of submatrices in certain positions requires rank deficiency elsewhere. Whereas, in the case of general matrices, rank deficiency of a large submatrix can imply rank deficiency of smaller, included submatrices, in the TN case rank deficiency of small submatrices can imply that of much larger ones. To be sure, in the generalm-by-ncase, a...

13. Chapter Eight Hadamard Products and Powers of TN Matrices
(pp. 167-184)

TheHadamard productof twom-by-nmatricesA= [aij] andB= [bij] is defined and denoted by$$A \circ B = [a_{ij} b_{ij}].$$Furthermore, ifAis am-by-nand entrywise nonnegative, thenA(t)denotes thetthHadamard powerofA, and is defined to be$$A^{(t)} = [a_{{ij}}^t],$$for anyt≥ 0.

Evidently, ifAandBare entrywise positive (that is, TP₁), thenABis also TP₁. Furthermore, with a bit more effort it is not difficult to demonstrate that ifAandBare TP₂ and 2-by-2, then$$\det{(A \circ B)} > 0,$$as both detAand detBare positive.

From the above simple...

14. Chapter Nine Extensions and Completions
(pp. 185-204)

The property that a matrix be TP is sufficiently strong that, at first glance, construction seems even more difficult than recognition. Of course, the elementary bidiagonal factorization provides an easy way simply to write down an example, but with this factorization it is very difficult to “design” many entries of the resulting matrix. Here, we review a variety of construction, extension and completion ideas for both TP and TN matrices.

Another way to produce a TP matrix is a technique that follows the simple diagram

This “exterior bordering” technique represents a strategy for extending any existing TP matrix by adding...

15. Chapter Ten Other Related Topics on TN Matrices
(pp. 205-218)

Part of the motivation for this work grew out of the fact that total positivity and, in particular, TP matrices arise in many parts of mathematics and have a broad history, and, therefore, a summary of this topic seems natural. This is what we have tried to accomplish here. There are many facts and properties about TP and TN matrices that neither fit well in another chapter nor are as lengthy as a separate chapter. Yet, these topics are worthy of being mentioned and discussed. Thus, in some sense this final chapter represents a catalog of additional topics dealing with...

16. Bibliography
(pp. 219-238)
17. List of Symbols
(pp. 239-244)
18. Index
(pp. 245-248)