Matrix analysis and entrywise positivity preservers / Apoorva Khare.

Format:

Book

Author/Creator:

Khare, Apoorva, author.

Series:

London Mathematical Society lecture note series ; 471.

London Mathematical Society lecture note series ; 471

Language:

English

Subjects (All):

Matrices.

Kernel functions.

Positive operators.

Composition operators.

Physical Description:

1 online resource (xxii, 292 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Cambridge : Cambridge University Press, 2022.

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.

Contents:

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

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.

Notes:

Title from publisher's bibliographic system (viewed on 30 Mar 2022).

ISBN:

1-108-85780-9

1-108-86712-X

OCLC:

1319466342