On Euclidean Distance Matrices and Spherical Configurations. A.Y. Alfakih Dept of Math and Statistics University of Windsor DIMACS Workshop on Optimization in Distance Geometry June 26-28, 2019 Outline I Survey of EDMs: I Characterizations. I Properties. I Classes of EDMs: Spherical and Nonspherical. I EDM Inverse Eigenvalue Problem. I Spherical Configurations I Yielding and Nonyielding Entries. I Unit Spherical EDMs which differ in 1 entry. I Two-Distance Sets. I The dimension of the affine span of the generating points of an EDM D is called the embedding dimension of D. I An EDM D is spherical if its generating points lie on a hypersphere. Otherwise, it is nonspherical. Definition 1 n I An n × n matrix D is an EDM if there exist points p ;:::; p in some Euclidean space such that: i j 2 dij = jjp − p jj for all i; j = 1;:::; n: I An EDM D is spherical if its generating points lie on a hypersphere. Otherwise, it is nonspherical. Definition 1 n I An n × n matrix D is an EDM if there exist points p ;:::; p in some Euclidean space such that: i j 2 dij = jjp − p jj for all i; j = 1;:::; n: I The dimension of the affine span of the generating points of an EDM D is called the embedding dimension of D. Definition 1 n I An n × n matrix D is an EDM if there exist points p ;:::; p in some Euclidean space such that: i j 2 dij = jjp − p jj for all i; j = 1;:::; n: I The dimension of the affine span of the generating points of an EDM D is called the embedding dimension of D. I An EDM D is spherical if its generating points lie on a hypersphere. Otherwise, it is nonspherical. y I w where Dw = e. Some times we set w = D e. T I s where e s = 1. Vector s fixes the origin. Two important choices: s = e=n and s = 2w. Important Vectors in EDM Theory n T I e the vector of all 1's in R and V : V e = 0 and T V V = In−1. T I s where e s = 1. Vector s fixes the origin. Two important choices: s = e=n and s = 2w. Important Vectors in EDM Theory n T I e the vector of all 1's in R and V : V e = 0 and T V V = In−1. y I w where Dw = e. Some times we set w = D e. Important Vectors in EDM Theory n T I e the vector of all 1's in R and V : V e = 0 and T V V = In−1. y I w where Dw = e. Some times we set w = D e. T I s where e s = 1. Vector s fixes the origin. Two important choices: s = e=n and s = 2w. T I Let e s = 1. This theorem can be re-stated as [Gower '85]: Let D be a real symmetric matrix with zero diagonal. Then D is EDM iff 1 T (D) = − (I − esT )D(I − seT ) 0: 2 Moreover, the embedding dimension of D = rank T (D). I B = T (D) is the Gram matrix of the generating points of D. Note that Bs = 0. I Given B, generating points of D are given by the rows of P, where B = PPT . Characterizing EDMs I Theorem [Schoenberg '35, Young and Householder '38]: Let D be a real symmetric matrix with zero diagonal. Then D is an EDM iff D is negative semidefinite on e?. I B = T (D) is the Gram matrix of the generating points of D. Note that Bs = 0. I Given B, generating points of D are given by the rows of P, where B = PPT . Characterizing EDMs I Theorem [Schoenberg '35, Young and Householder '38]: Let D be a real symmetric matrix with zero diagonal. Then D is an EDM iff D is negative semidefinite on e?. T I Let e s = 1. This theorem can be re-stated as [Gower '85]: Let D be a real symmetric matrix with zero diagonal. Then D is EDM iff 1 T (D) = − (I − esT )D(I − seT ) 0: 2 Moreover, the embedding dimension of D = rank T (D). I Given B, generating points of D are given by the rows of P, where B = PPT . Characterizing EDMs I Theorem [Schoenberg '35, Young and Householder '38]: Let D be a real symmetric matrix with zero diagonal. Then D is an EDM iff D is negative semidefinite on e?. T I Let e s = 1. This theorem can be re-stated as [Gower '85]: Let D be a real symmetric matrix with zero diagonal. Then D is EDM iff 1 T (D) = − (I − esT )D(I − seT ) 0: 2 Moreover, the embedding dimension of D = rank T (D). I B = T (D) is the Gram matrix of the generating points of D. Note that Bs = 0. Characterizing EDMs I Theorem [Schoenberg '35, Young and Householder '38]: Let D be a real symmetric matrix with zero diagonal. Then D is an EDM iff D is negative semidefinite on e?. T I Let e s = 1. This theorem can be re-stated as [Gower '85]: Let D be a real symmetric matrix with zero diagonal. Then D is EDM iff 1 T (D) = − (I − esT )D(I − seT ) 0: 2 Moreover, the embedding dimension of D = rank T (D). I B = T (D) is the Gram matrix of the generating points of D. Note that Bs = 0. I Given B, generating points of D are given by the rows of P, where B = PPT . n n 1 T T T : Sh ! Ss : T (D) = − 2 (I − es )D(I − se ) I n n T T K : Ss ! Sh : K(B) = diag(B)e + e(diag(B)) − 2B. I Theorem [Critchley '88 ]: −1 −1 T j n = (Kj n ) and Kj n = (T j n ) : Sh Ss Ss Sh i j 2 T I dij = jjp − p jj = Bii + Bjj − 2Bij , where B = PP . Thus n D = K(B) and D 2 Sh is an EDM iff B = T (D) 0. Proof Define: n Ss = fA : A is sym; As = 0g I n Sh = fA : A is sym; diag(A) = 0g. I Theorem [Critchley '88 ]: −1 −1 T j n = (Kj n ) and Kj n = (T j n ) : Sh Ss Ss Sh i j 2 T I dij = jjp − p jj = Bii + Bjj − 2Bij , where B = PP . Thus n D = K(B) and D 2 Sh is an EDM iff B = T (D) 0. Proof Define: n Ss = fA : A is sym; As = 0g I n Sh = fA : A is sym; diag(A) = 0g. n n 1 T T T : Sh ! Ss : T (D) = − 2 (I − es )D(I − se ) I n n T T K : Ss ! Sh : K(B) = diag(B)e + e(diag(B)) − 2B. i j 2 T I dij = jjp − p jj = Bii + Bjj − 2Bij , where B = PP . Thus n D = K(B) and D 2 Sh is an EDM iff B = T (D) 0. Proof Define: n Ss = fA : A is sym; As = 0g I n Sh = fA : A is sym; diag(A) = 0g. n n 1 T T T : Sh ! Ss : T (D) = − 2 (I − es )D(I − se ) I n n T T K : Ss ! Sh : K(B) = diag(B)e + e(diag(B)) − 2B. I Theorem [Critchley '88 ]: −1 −1 T j n = (Kj n ) and Kj n = (T j n ) : Sh Ss Ss Sh Proof Define: n Ss = fA : A is sym; As = 0g I n Sh = fA : A is sym; diag(A) = 0g. n n 1 T T T : Sh ! Ss : T (D) = − 2 (I − es )D(I − se ) I n n T T K : Ss ! Sh : K(B) = diag(B)e + e(diag(B)) − 2B. I Theorem [Critchley '88 ]: −1 −1 T j n = (Kj n ) and Kj n = (T j n ) : Sh Ss Ss Sh i j 2 T I dij = jjp − p jj = Bii + Bjj − 2Bij , where B = PP . Thus n D = K(B) and D 2 Sh is an EDM iff B = T (D) 0. T T T I J = VV , where V e = 0 and V V = In−1. I F = fB 0 : Be = 0g is a face of the PSD cone. F = fB = VXV T ; X 0g is isomorphic to PSD cone of order n − 1. T I X = V BV is called the projected Gram matrix. Moreover, X 0 and of rank r iff B 0 and of rank r. I Define [A. , Khandani and Wolkowicz '99]: T KV (X ) = K(VXV ) and T T TV (D) = V T (D)V = −V DV =2. Then the cone of EDMs of order n is the image of the PSD cone of order n − 1 under KV . Projected Gram Matrices T I Set s = e=n and let J = I − ee =n. Hence B = −JDJ=2 and Be = 0. I F = fB 0 : Be = 0g is a face of the PSD cone. F = fB = VXV T ; X 0g is isomorphic to PSD cone of order n − 1. T I X = V BV is called the projected Gram matrix. Moreover, X 0 and of rank r iff B 0 and of rank r. I Define [A. , Khandani and Wolkowicz '99]: T KV (X ) = K(VXV ) and T T TV (D) = V T (D)V = −V DV =2. Then the cone of EDMs of order n is the image of the PSD cone of order n − 1 under KV .