Schoenberg: from metric geometry to matrix positivity Apoorva Khare Indian Institute of Science Eigenfunctions Seminar (with Gautam Bharali) IISc , April 2019
Entrywise functions preserving positivity Definitions: A real symmetric matrix A n × n is positive semidefinite if its quadratic 1 form is so: x T Ax ≥ 0 for all x ∈ R n . (Hence σ ( A ) ⊂ [0 , ∞ ) .) Given n ≥ 1 and I ⊂ R , let P n ( I ) denote the n × n positive 2 (semidefinite) matrices , with entries in I . (Say P n = P n ( R ) .) Apoorva Khare , IISc Bangalore 2 / 14
Entrywise functions preserving positivity Definitions: A real symmetric matrix A n × n is positive semidefinite if its quadratic 1 form is so: x T Ax ≥ 0 for all x ∈ R n . (Hence σ ( A ) ⊂ [0 , ∞ ) .) Given n ≥ 1 and I ⊂ R , let P n ( I ) denote the n × n positive 2 (semidefinite) matrices , with entries in I . (Say P n = P n ( R ) .) A function f : I → R acts entrywise on a matrix A ∈ I n × n via: 3 f [ A ] := ( f ( a jk )) n j,k =1 . Apoorva Khare , IISc Bangalore 2 / 14
Entrywise functions preserving positivity Definitions: A real symmetric matrix A n × n is positive semidefinite if its quadratic 1 form is so: x T Ax ≥ 0 for all x ∈ R n . (Hence σ ( A ) ⊂ [0 , ∞ ) .) Given n ≥ 1 and I ⊂ R , let P n ( I ) denote the n × n positive 2 (semidefinite) matrices , with entries in I . (Say P n = P n ( R ) .) A function f : I → R acts entrywise on a matrix A ∈ I n × n via: 3 f [ A ] := ( f ( a jk )) n j,k =1 . Problem: For which functions f : I → R is it true that f [ A ] ∈ P n for all A ∈ P n ( I )? Apoorva Khare , IISc Bangalore 2 / 14
Entrywise functions preserving positivity Definitions: A real symmetric matrix A n × n is positive semidefinite if its quadratic 1 form is so: x T Ax ≥ 0 for all x ∈ R n . (Hence σ ( A ) ⊂ [0 , ∞ ) .) Given n ≥ 1 and I ⊂ R , let P n ( I ) denote the n × n positive 2 (semidefinite) matrices , with entries in I . (Say P n = P n ( R ) .) A function f : I → R acts entrywise on a matrix A ∈ I n × n via: 3 f [ A ] := ( f ( a jk )) n j,k =1 . Problem: For which functions f : I → R is it true that f [ A ] ∈ P n for all A ∈ P n ( I )? (Long history!) The Schur Product Theorem [Schur , Crelle 1911] says: If A, B ∈ P n , then so is A ◦ B := ( a jk b jk ) . Apoorva Khare , IISc Bangalore 2 / 14
Entrywise functions preserving positivity Definitions: A real symmetric matrix A n × n is positive semidefinite if its quadratic 1 form is so: x T Ax ≥ 0 for all x ∈ R n . (Hence σ ( A ) ⊂ [0 , ∞ ) .) Given n ≥ 1 and I ⊂ R , let P n ( I ) denote the n × n positive 2 (semidefinite) matrices , with entries in I . (Say P n = P n ( R ) .) A function f : I → R acts entrywise on a matrix A ∈ I n × n via: 3 f [ A ] := ( f ( a jk )) n j,k =1 . Problem: For which functions f : I → R is it true that f [ A ] ∈ P n for all A ∈ P n ( I )? (Long history!) The Schur Product Theorem [Schur , Crelle 1911] says: If A, B ∈ P n , then so is A ◦ B := ( a jk b jk ) . As a consequence , f ( x ) = x k ( k ≥ 0 ) preserves positivity on P n for all n . Apoorva Khare , IISc Bangalore 2 / 14
Entrywise functions preserving positivity Definitions: A real symmetric matrix A n × n is positive semidefinite if its quadratic 1 form is so: x T Ax ≥ 0 for all x ∈ R n . (Hence σ ( A ) ⊂ [0 , ∞ ) .) Given n ≥ 1 and I ⊂ R , let P n ( I ) denote the n × n positive 2 (semidefinite) matrices , with entries in I . (Say P n = P n ( R ) .) A function f : I → R acts entrywise on a matrix A ∈ I n × n via: 3 f [ A ] := ( f ( a jk )) n j,k =1 . Problem: For which functions f : I → R is it true that f [ A ] ∈ P n for all A ∈ P n ( I )? (Long history!) The Schur Product Theorem [Schur , Crelle 1911] says: If A, B ∈ P n , then so is A ◦ B := ( a jk b jk ) . As a consequence , f ( x ) = x k ( k ≥ 0 ) preserves positivity on P n for all n . k =0 c k x k is (Pólya–Szegö , 1925): Taking sums and limits , if f ( x ) = � ∞ convergent and c k ≥ 0 , then f [ − ] preserves positivity. Apoorva Khare , IISc Bangalore 2 / 14
Entrywise functions preserving positivity Definitions: A real symmetric matrix A n × n is positive semidefinite if its quadratic 1 form is so: x T Ax ≥ 0 for all x ∈ R n . (Hence σ ( A ) ⊂ [0 , ∞ ) .) Given n ≥ 1 and I ⊂ R , let P n ( I ) denote the n × n positive 2 (semidefinite) matrices , with entries in I . (Say P n = P n ( R ) .) A function f : I → R acts entrywise on a matrix A ∈ I n × n via: 3 f [ A ] := ( f ( a jk )) n j,k =1 . Problem: For which functions f : I → R is it true that f [ A ] ∈ P n for all A ∈ P n ( I )? (Long history!) The Schur Product Theorem [Schur , Crelle 1911] says: If A, B ∈ P n , then so is A ◦ B := ( a jk b jk ) . As a consequence , f ( x ) = x k ( k ≥ 0 ) preserves positivity on P n for all n . k =0 c k x k is (Pólya–Szegö , 1925): Taking sums and limits , if f ( x ) = � ∞ convergent and c k ≥ 0 , then f [ − ] preserves positivity. Question: Anything else? Apoorva Khare , IISc Bangalore 2 / 14
Schoenberg’s theorem Interestingly , the answer is no , if we want to preserve positivity in all dimensions: Apoorva Khare , IISc Bangalore 3 / 14
Schoenberg’s theorem Interestingly , the answer is no , if we want to preserve positivity in all dimensions: Theorem (Schoenberg , Duke Math. J. 1942 ; Rudin , Duke Math. J. 1959) Suppose I = ( − 1 , 1) and f : I → R . The following are equivalent: f [ A ] ∈ P n for all A ∈ P n ( I ) and all n ≥ 1 . 1 f is analytic on I and has nonnegative Taylor coefficients. 2 k =0 c k x k on ( − 1 , 1) with all c k ≥ 0 . In other words , f ( x ) = � ∞ Apoorva Khare , IISc Bangalore 3 / 14
Schoenberg’s theorem Interestingly , the answer is no , if we want to preserve positivity in all dimensions: Theorem (Schoenberg , Duke Math. J. 1942 ; Rudin , Duke Math. J. 1959) Suppose I = ( − 1 , 1) and f : I → R . The following are equivalent: f [ A ] ∈ P n for all A ∈ P n ( I ) and all n ≥ 1 . 1 f is analytic on I and has nonnegative Taylor coefficients. 2 k =0 c k x k on ( − 1 , 1) with all c k ≥ 0 . In other words , f ( x ) = � ∞ Schoenberg’s result is the (harder) converse to that of his advisor: Schur. Vasudeva (1979) proved a variant , over I = (0 , ∞ ) . Apoorva Khare , IISc Bangalore 3 / 14
Schoenberg’s theorem Interestingly , the answer is no , if we want to preserve positivity in all dimensions: Theorem (Schoenberg , Duke Math. J. 1942 ; Rudin , Duke Math. J. 1959) Suppose I = ( − 1 , 1) and f : I → R . The following are equivalent: f [ A ] ∈ P n for all A ∈ P n ( I ) and all n ≥ 1 . 1 f is analytic on I and has nonnegative Taylor coefficients. 2 k =0 c k x k on ( − 1 , 1) with all c k ≥ 0 . In other words , f ( x ) = � ∞ Schoenberg’s result is the (harder) converse to that of his advisor: Schur. Vasudeva (1979) proved a variant , over I = (0 , ∞ ) . Upshot: Preserving positivity in all dimensions is a rigid condition � implies real analyticity, absolute monotonicity. . . Apoorva Khare , IISc Bangalore 3 / 14
Schoenberg’s motivations: metric geometry Endomorphisms of matrix spaces with positivity constraints related to: matrix monotone functions (Loewner) preservers of matrix properties (rank , inertia , . . . ) real-stable/hyperbolic polynomials (Borcea , Branden , Liggett , Marcus , Spielman , Srivastava. . . ) positive definite functions (von Neumann , Bochner , Schoenberg . . . ) Apoorva Khare , IISc Bangalore 4 / 14
Schoenberg’s motivations: metric geometry Endomorphisms of matrix spaces with positivity constraints related to: matrix monotone functions (Loewner) preservers of matrix properties (rank , inertia , . . . ) real-stable/hyperbolic polynomials (Borcea , Branden , Liggett , Marcus , Spielman , Srivastava. . . ) positive definite functions (von Neumann , Bochner , Schoenberg . . . ) Definition f : [0 , ∞ ) → R is positive definite on a metric space ( X, d ) if [ f ( d ( x j , x k ))] n j,k =1 ∈ P n , for all n ≥ 1 and all x 1 , . . . , x n ∈ X . Apoorva Khare , IISc Bangalore 4 / 14
Schoenberg’s motivations: metric geometry Endomorphisms of matrix spaces with positivity constraints related to: matrix monotone functions (Loewner) preservers of matrix properties (rank , inertia , . . . ) real-stable/hyperbolic polynomials (Borcea , Branden , Liggett , Marcus , Spielman , Srivastava. . . ) positive definite functions (von Neumann , Bochner , Schoenberg . . . ) Definition f : [0 , ∞ ) → R is positive definite on a metric space ( X, d ) if [ f ( d ( x j , x k ))] n j,k =1 ∈ P n , for all n ≥ 1 and all x 1 , . . . , x n ∈ X . Plan for rest of the talk: Discuss the path from metric geometry , through positive definite functions , to Schoenberg’s theorem. Apoorva Khare , IISc Bangalore 4 / 14
Distance geometry How did the study of positivity and its preservers begin? Apoorva Khare , IISc Bangalore 5 / 14
Distance geometry How did the study of positivity and its preservers begin? In the 1900s , the notion of a metric space emerged from the works of Fréchet and Hausdorff. . . Now ubiquitous in science (mathematics , physics , economics , statistics , computer science. . . ). Apoorva Khare , IISc Bangalore 5 / 14
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