Matrix Analysis and Entrywise Positivity Preservers

These self-contained lecture notes develop classical and modern results on positive matrices and kernels, and on transforms preserving them.

Apoorva Khare (Author)

9781108792042, Cambridge University Press

Paperback / softback, published 31 March 2022

300 pages
22.6 x 15.2 x 1.5 cm, 0.51 kg

'The opening notes of this symphony of ideas were written by Schur in 1911. Schoenberg, Loewner, Rudin, Herz, Hiai, FitzGerald, Jain, Guillot, Rajaratnam, Belton, Putinar, and others composed new themes and variations. Now, Khare has orchestrated a masterwork that includes his own harmonies in an elegant synthesis. This is a work of impressive scholarship.' Roger Horn, University of Utah, Retired

Matrices and kernels with positivity structures, and the question of entrywise functions preserving them, have been studied throughout the 20th century, attracting recent interest in connection to high-dimensional covariance estimation. This is the first book to systematically develop the theoretical foundations of the entrywise calculus, focusing on entrywise operations - or transforms - of matrices and kernels with additional structure, which preserve positive semidefiniteness. Designed as an introduction for students, it presents an in-depth and comprehensive view of the subject, from early results to recent progress. Topics include: structural results about, and classifying the preservers of positive semidefiniteness and other Loewner properties (monotonicity, convexity, super-additivity); historical connections to metric geometry; classical connections to moment problems; and recent connections to combinatorics and Schur polynomials. Based on the author's course, the book is structured for use as lecture notes, including exercises for students, yet can also function as a comprehensive reference text for experts.

Part I. Preliminaries, Entrywise Powers Preserving Positivity in Fixed Dimension: 1. The cone of positive semidefinite matrices
2. The Schur product theorem and nonzero lower bounds
3. Totally positive (TP) and totally non-negative (TN) matrices
4. TP matrices – generalized Vandermonde and Hankel moment matrices
5. Entrywise powers preserving positivity in fixed dimension
6. Mid-convex implies continuous, and 2 x 2 preservers
7. Entrywise preservers of positivity on matrices with zero patterns
8. Entrywise powers preserving positivity, monotonicity, superadditivity
9. Loewner convexity and single matrix encoders of preservers
10. Exercises
Part II. Entrywise Functions Preserving Positivity in All Dimensions: 11. History – Shoenberg, Rudin, Vasudeva, and metric geometry
12. Loewner's determinant calculation in Horn's thesis
13. The stronger Horn–Loewner theorem, via mollifiers
14. Stronger Vasudeva and Schoenberg theorems, via Bernstein's theorem
15. Proof of stronger Schoenberg Theorem (Part I) – positivity certificates
16. Proof of stronger Schoenberg Theorem (Part II) – real analyticity
17. Proof of stronger Schoenberg Theorem (Part III) – complex analysis
18. Preservers of Loewner positivity on kernels
19. Preservers of Loewner monotonicity and convexity on kernels
20. Functions acting outside forbidden diagonal blocks
21. The Boas–Widder theorem on functions with positive differences
22. Menger's results and Euclidean distance geometry
23. Exercises
Part III. Entrywise Polynomials Preserving Positivity in Fixed Dimension: 24. Entrywise polynomial preservers and Horn–Loewner type conditions
25. Polynomial preservers for rank-one matrices, via Schur polynomials
26. First-order approximation and leading term of Schur polynomials
27. Exact quantitative bound – monotonicity of Schur ratios
28. Polynomial preservers on matrices with real or complex entries
29. Cauchy and Littlewood's definitions of Schur polynomials
30. Exercises.

Subject Areas: Algebra [PBF]