Matrix analysis and entrywise positivity preservers Apoorva Khare.

Includes bibliographical references and index.

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.