Matrix Analysis and Entrywise Positivity Preservers

ISBN-13: 9781108792042 / Angielski / Miękka / 2022 / 300 str.

Bangalore) Khare Apoorva (Indian Institute of Science

Opis

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

Recenzje

'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

Spis treści

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.