The cone of positive semidefinite matrices -- The Schur product theorem and nonzero lower bounds -- Totally positive (T P) and Totally non-negative (T N) matrices -- T P matrices -- generalized Vandermonde and Hankel moment matrices -- Entrywise powers preserving positivity in fixed dimension -- Mid-convex implies continuous, and 2 x 2 preservers -- Entrywise preservers of positivity on matrices with zero patterns -- Entrywise powers preserving positivity, monotonicity, superadditivity -- Loewner convexity and single matrix encoders of preservers -- Exercises -- History -- Schoenberg, Rudin, Vasudeva, and metric geometry -- Loewner's determinant calculation in Horn's thesis -- The stronger Horn-Loewner theorem, via mollifiers -- Stronger Vasudeva and Schoenberg theorems, via Bernstein's theorem -- Proof of stronger Schoenberg theorem (part I) -- positivity certificates -- Proof of stronger Schoenberg theorem (part II) -- real analyticity -- Proof of stronger Schoenberg theorem (part III) -- complex analysis -- Preservers of Loewner positivity on kernels -- Preservers of Loewner monotonicity and convexity on kernels -- Functions acting outside forbidden diagonal blocks -- The Boas-Widder theorem on functions with positive differences -- Menger's results and Euclidean distance geometry -- Exercises -- Entrywise polynomial preservers and Horn-Loewner type conditions -- Polynomial preservers for rank-one matrices, via Schur polynomials -- First-order approximation and leading term of Schur polynomials -- Exact quantitative bound -- monotonicity of Schur ratios -- Polynomial preservers on matrices with real or complex entries -- Cauchy and Littlewood's definitions of Schur polynomials -- Exercises.
Summary:
"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."-- Publisher's website.
Series:
London Mathematical Society lecture note series ; 471
This resource is supported by the Institute of Museum and Library Services under the provisions of the Library Services and Technology Act as administered by State Library of Iowa.