Matrix positivity preservers in fixed dimension. I
Copyright policy )Matrix positivity preservers in fixed dimension. I
A classical theorem of I.J. Schoenberg characterizes functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size. Obtaining similar characterizations in fixed dimension is intricate. In this note, we provide a solution to this problem in the polynomial case. As consequences, we derive tight linear matrix inequalities for Hadamard powers of positive semidefinite matrices, and a sharp asymptotic bound for the matrix cube problem involving Hadamard powers.
- University of Delaware United States
- Newcastle University United Kingdom
- University of California, Santa Barbara United States
- Stanford University United States
- Lancaster University United Kingdom
positive definite matrix, pencils of matrices, Hadamard product, Schur polynomial, variational inequality, 510, characteristic value, Positive matrices and their generalizations; cones of matrices, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Mathematics - Combinatorics, linear matrix inequalities, 15A45 (primary), 05E05, 15A12, 15B48, 26C05, 62H99, 65F15 (secondary), Hadamard powers, spectrahedron, positivity preservers, positive semidefinite matrices, matrix positivity preservers, entrywise function, Variational inequalities, 004, Functional Analysis (math.FA), Mathematics - Functional Analysis, Miscellaneous inequalities involving matrices, Mathematics - Classical Analysis and ODEs, correlation matrix, Linear preserver problems, Combinatorics (math.CO), Matrix pencils, linear matrix inequality
positive definite matrix, pencils of matrices, Hadamard product, Schur polynomial, variational inequality, 510, characteristic value, Positive matrices and their generalizations; cones of matrices, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Mathematics - Combinatorics, linear matrix inequalities, 15A45 (primary), 05E05, 15A12, 15B48, 26C05, 62H99, 65F15 (secondary), Hadamard powers, spectrahedron, positivity preservers, positive semidefinite matrices, matrix positivity preservers, entrywise function, Variational inequalities, 004, Functional Analysis (math.FA), Mathematics - Functional Analysis, Miscellaneous inequalities involving matrices, Mathematics - Classical Analysis and ODEs, correlation matrix, Linear preserver problems, Combinatorics (math.CO), Matrix pencils, linear matrix inequality
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