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On Euclidean Distance Matrices and Spherical Configurations 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 .
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