Preservers for classes of positive matrices Alexander Belton Department of Mathematics and Statistics Lancaster University, United Kingdom a.belton@lancaster.ac.uk 18th Workshop: Noncommutative Probability, Operator Algebras, Random Matrices and Related Topics, with Applications B¸ edlewo 20th July 2018
Structure of the talk Joint work with Dominique Guillot (Delaware), Apoorva Khare (IISc Bangalore) and Mihai Putinar (UCSB and Newcastle) Two products on matrices 1 Positive definiteness 2 Theorems of Schur, Schoenberg and Horn 3 Positivity preservation 4 For fixed dimension 1 For moment matrices 2 For totally positive matrices 3 Alexander Belton (Lancaster University) Positivity preservers MRCC B¸ edlewo, 20vii18 2 / 23
Two matrix products Notation The set of n × n matrices with entries in a set K ⊆ C is denoted M n ( K ). Products The vector space M n ( C ) is an associative algebra for at least two different products: if A = ( a ij ) and B = ( b ij ) then n � ( AB ) ij := a ik b kj (standard) k =1 and ( A ◦ B ) ij := a ij b ij (Hadamard) . Alexander Belton (Lancaster University) Positivity preservers MRCC B¸ edlewo, 20vii18 3 / 23
Positive definiteness Definition A matrix A ∈ M n ( C ) is positive semidefinite if n x ∗ A x = � for all x ∈ C n . x i a ij x j � 0 i , j =1 A matrix A ∈ M n ( C ) is positive definite if n for all x ∈ C n \ { 0 } . x ∗ A x = � x i a ij x j > 0 i , j =1 Remark The subset of M n ( K ) consisting of positive semidefinite matrices is denoted M n ( K ) + .The set M n ( C ) + is a cone : closed under sums and under multiplication by elements of R + . Alexander Belton (Lancaster University) Positivity preservers MRCC B¸ edlewo, 20vii18 4 / 23
Some consequences Symmetry If A ∈ M n ( K ) + then A T ∈ M n ( K ) + : note that 0 � ( x ∗ A x ) T = y ∗ A T y for all y = x ∈ C n . Hermitianity If A ∈ M n ( C ) + then A = A ∗ : note that x ∗ A ∗ x = ( x ∗ A x ) ∗ = x ∗ A x A = A ∗ ( x ∈ C n ) = ⇒ (polarisation) . Non-negative eigenvalues If A ∈ M n ( C ) + = U ∗ diag ( λ 1 , . . . , λ n ) U , where U ∈ M n ( C ) is unitary, then λ i = ( U ∗ e i ) ∗ A ( U ∗ e i ) � 0. Conversely, if A ∈ M n ( C ) is Hermitian and has non-negative eigenvalues � √ λ 1 , . . . , √ λ n then A = B ∗ B ∈ M n ( C ) + , where B = A 1 / 2 = U ∗ diag � U . Alexander Belton (Lancaster University) Positivity preservers MRCC B¸ edlewo, 20vii18 5 / 23
The Schur product theorem Theorem 1 (Schur, 1911) If A, B ∈ M n ( C ) + then A ◦ B ∈ M n ( C ) + . Proof. x ∗ ( A ◦ B ) x tr ( diag ( x ) ∗ B diag ( x ) A T � = ( A T ) 1 / 2 diag ( x ) ∗ B diag ( x )( A T ) 1 / 2 � � = . tr Alexander Belton (Lancaster University) Positivity preservers MRCC B¸ edlewo, 20vii18 6 / 23
A question of P´ olya and Szeg¨ o Corollary 2 n =0 a n z n is analytic on K and a n � 0 for all n � 0 then If f ( z ) = � ∞ � � f [ A ] := f ( a ij ) ∈ M n ( C ) + for all A = ( a ij ) ∈ M n ( K ) + and all n � 1 . Observation The functions in the corollary preserve positivity on M n ( K ) + regardless of the dimension n . Question (P´ olya–Szeg¨ o, 1925) Are there any other functions with this property? Alexander Belton (Lancaster University) Positivity preservers MRCC B¸ edlewo, 20vii18 7 / 23
Schoenberg’s theorem Theorem 3 (Schoenberg, 1942) If f : [ − 1 , 1] → R is (i) continuous and � � (ii) such that f [ A ] ∈ M n ( R ) + for all A ∈ M n [ − 1 , 1] + and all n � 1 then f is absolutely monotonic: ∞ � a n x n f ( x ) = for all x ∈ [ − 1 , 1] , where a n � 0 for all n � 0 . n =0 Proof (Christensen–Ressel, 1978). � � { f : [ − 1 , 1] → R | f (1) = 1 , f [ A ] ∈ M n ( R ) + ∀ A ∈ M n [ − 1 , 1] + , n � 1 } is a Choquet simplex with closed set of extreme points { x n : n � 0 } ∪ { 1 { 1 } − 1 {− 1 } , 1 {− 1 , 1 } } . Alexander Belton (Lancaster University) Positivity preservers MRCC B¸ edlewo, 20vii18 8 / 23
Horn’s theorem Theorem 4 (Horn, 1969) If f : (0 , ∞ ) → R is continuous and such that f [ A ] ∈ M n ( R ) + for all � � A ∈ M n (0 , ∞ ) + , where n � 3 , then (i) f ∈ C n − 3 � � (0 , ∞ ) and (ii) f ( k ) ( x ) � 0 for all k = 0 , . . . , n − 3 and all x > 0 . If, further, f ∈ C n − 1 � � , then f ( k ) ( x ) � 0 for all k = 0 , . . . , n − 1 (0 , ∞ ) and all x > 0 . Lemma 5 If f : D (0 , ρ ) → R is analytic, where ρ > 0 , and such that f [ A ] ∈ M n ( R ) + � � whenever A ∈ M n (0 , ρ ) + has rank at most one, then the first n non-zero Taylor coefficients of f are strictly positive. Alexander Belton (Lancaster University) Positivity preservers MRCC B¸ edlewo, 20vii18 9 / 23
A theorem for fixed dimension Theorem 6 (B–G–K–P, 2015) Let ρ > 0 , n � 1 and n − 1 c j z j + c ′ z m , � f ( z ) = j =0 where c = ( c 0 , . . . , c n − 1 ) ∈ R n , c ′ ∈ R and m � 0 .The following are equivalent. � � (i) f [ A ] ∈ M n ( C ) + for all A ∈ M n D (0 , ρ ) + . + and c ′ ∈ R + , or c ∈ (0 , ∞ ) n and c ′ � − C ( c ; m , ρ ) − 1 , (ii) Either c ∈ R n where � 2 ρ m − j n − 1 � 2 � m − j − 1 � m � C ( c ; m , ρ ) := . j n − j − 1 c j j =0 � � (iii) f [ A ] ∈ M n ( R ) + for all A ∈ M n (0 , ρ ) + with rank at most one. Alexander Belton (Lancaster University) Positivity preservers MRCC B¸ edlewo, 20vii18 10 / 23
A determinant identity with Schur polynomials For all t ∈ R , n � 1 and c = ( c 0 , . . . , c n − 1 ) ∈ F n , let the polynomial p ( z ; t , c , m ) := t ( c 0 + · · · + c n − 1 z n − 1 ) − z m ( z ∈ C ) . Given a non-increasing n -tuple k = ( k n � · · · � k 1 ) ∈ Z n + , let σ k ( x 1 , . . . , x n ) := det( x k j + j − 1 ) V n ( x 1 , . . . , x n ) := det( x j − 1 i and ) . det( x j − 1 i ) i Theorem 7 Let m � n � 1 . If µ ( m , n , j ) := ( m − n + 1 , 1 , . . . , 1 , 0 , . . . , 0) , where there are n − j − 1 ones and j zeros, and c ∈ ( F × ) n then, for all u , v ∈ F n , n − 1 � n − 1 σ µ ( m , n , j ) ( u ) σ µ ( m , n , j ) ( v ) det p [ uv T ; t , c , m ]= t n − 1 V n ( u ) V n ( v ) � � � t − c j . c j j =0 j =0 Alexander Belton (Lancaster University) Positivity preservers MRCC B¸ edlewo, 20vii18 11 / 23
Proof that (ii) = ⇒ (i) in Theorem 6 when m � n By the Schur product theorem, it suffices to assume that − C ( c ; m , ρ ) − 1 � c ′ < 0 < c 0 , . . . , c n − 1 . � � The key step is to show that f [ A ] ∈ M n ( C ) + if A ∈ M n D (0 , ρ ) + has rank at most one. Given this, the proof concludes by induction. The n = 1 case is immediate, since every 1 × 1 matrix has rank at most one. Assume the n − 1 case holds. Since n − 1 c j z j + c ′ z m � | c ′ | | c ′ | − 1 ( c 0 + · · · + c n − 1 z n − 1 ) − z m � � f ( z ) = = j =0 | c ′ | p ( z ; | c ′ | − 1 , c , m ) , = it suffices to show p [ A ] := p [ A ; | c ′ | − 1 , c , m ] ∈ M n ( C ) + if A ∈ M n ( C ) + . Alexander Belton (Lancaster University) Positivity preservers MRCC B¸ edlewo, 20vii18 12 / 23
Two lemmas Lemma 8 (FitzGerald–Horn, 1977) Given A = ( a ij ) ∈ M n ( C ) + , set z ∈ C n to equal ( a in / √ a nn ) if a nn � = 0 and to be the zero vector otherwise.Then A − zz ∗ ∈ M n ( C ) + and the final row and column of this matrix are zero. + then zz ∗ ∈ M n � � � � If A ∈ M n D (0 , ρ ) D (0 , ρ ) + , since � a ii a in � � � = a ii a nn − | a in | 2 � 0 . � � a in a nn Lemma 9 If F : C → C is differentiable then � 1 ( z − w ) F ′ ( λ z + (1 − λ ) w ) d λ F ( z ) = F ( w ) + ( z , w ∈ C ) . 0 Alexander Belton (Lancaster University) Positivity preservers MRCC B¸ edlewo, 20vii18 13 / 23
Proof that (ii) = ⇒ (i) continued By Lemmas 8 and 9, � 1 p [ A ] = p [ zz ∗ ] + ( A − zz ∗ ) ◦ p ′ [ λ A + (1 − λ ) zz ∗ ] d λ. 0 Now, p ′ ( z ; | c ′ | − 1 , c , m ) = m p ( z ; | c ′ | − 1 / m , ( c 1 , . . . , ( n − 1) c n − 1 ) , m − 1) so the integrand is positive semidefinite, by the inductive assumption, since � m C ( c 1 , . . . , ( n − 1) c n − 1 ); m − 1 , ρ ) � C ( c ; m , ρ ) . Hence p [ A ] is positive semidefinite, as required. Alexander Belton (Lancaster University) Positivity preservers MRCC B¸ edlewo, 20vii18 14 / 23
Hankel matrices Hankel matrices Let µ be a measure on R with moments of all orders, and let � x n µ ( d x ) s n = s n ( µ ) := ( n � 0) . R The Hankel matrix associated with µ is s 0 s 1 s 2 . . . s 1 s 2 s 3 . . . H µ := = ( s i + j ) i , j � 0 . s 2 s 3 s 4 . . . . . . ... . . . . . . Alexander Belton (Lancaster University) Positivity preservers MRCC B¸ edlewo, 20vii18 15 / 23
The Hamburger moment problem Theorem 10 (Hamburger) A sequence ( s n ) n � 0 is the moment sequence for a positive Borel measure on R if and only if the associated Hankel matrix is positive semidefinite. Corollary 11 A map f preserves positivity when applied entrywise to Hankel matrices if and only if it maps moment sequences to themselves: given any positive Borel measure µ , � � f s n ( µ ) = s n ( ν ) ( n � 0) for some positive Borel measure ν . Alexander Belton (Lancaster University) Positivity preservers MRCC B¸ edlewo, 20vii18 16 / 23
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