Lorentz space as a setting for metric embeddability
Citation for published version (APA):Seidel, J. J. (1985). Lorentz space as a setting for metric embeddability. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8506). Technische Hogeschool Eindhoven.
Document status and date: Published: 01/01/1985
Document Version:
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:
www.tue.nl/taverne
Take down policy
If you believe that this document breaches copyright please contact us at:
openaccess@tue.nl
providing details and we will investigate your claim.
EINDHOVEN UNIVERSITY OF TECHNOLOGY
Department of Mathematics and Computing Science
r
Memorandum 1985-06
Lorentz space as a setting for metric embeddability by J.J. Seidel
L
April 1985 University of TechnologyDepartment of Mathematics and Computing Science PO Box 513, Eindhoven
The Netherlands
Lorentz space as a setting for metric
embeddability·)
J.J. Seidel
1. Introduction
A distance matrix D is a symmetric matrix having positive entries apart from the zero diagonal. Its entrees may be viewed as the distances of a finite semimetric space. We will be interested in the special distance matrices for which the corresponding semimetric space is isometrically embeddable in the
n
hypercube. embeddable in ~ with the ll-metric p~y) = ~ IXi -Yil' hyper-- (=1
metric. and of negative type. respectively. There are strict implications between these spaces (cf. M. Deza [4]. J.B. Kelly [6]. Assouad-Deza [1]):
Cube embeddable =;> lI-embeddable =;> hypergeometric =;> negative type. Since the corresponding distance matrices all have one positive eigenvalue, we expose them in the setting of Lorentz space R1•n right from the start.
2. Definitions
Lorentz space Rl.n is real space of dimension n
+
1 provided with an inner product of signature (l.n). With respect to a standard orthonormal basis §o'~I' .... ~ with ~.~)=
1 for i=
O. and=
-1 for i= 1 •...•
n. the inner product for IG=
~ ~i.!k.JI.
=
~ 1Ji jk readsThe norm of a vector is the inner product of the vector with itself. The
lio
ht cone C is the set of the isotropic vectors, that is. of the vectors of norm zero. Theorem 2.1. Any distance matrix D of size n+
1 having one positiveeigen-value, may be viewed as the Gram matrix of the inner products of n
+
1 isotropic vectors in W·n .• ) This note was written while the author was visiting the MEHTA RESEARCH INSTITUTE,Allahabad. India. January 1985.
This follows from diagonalization of D. Given D as in the theorem, we denote the isotropic vectors by £0'.£1' ..•• .!:n E: R1.n.
Clearly, the vector
is inside, on, or outside the light cone C. whenever its norm
is positive, zero, or negative. respectively. A vector
(!,
+
r',+ .., +
c.: - c.: _ . , . - c.:.::.\1 ~2 ~ ""'t:+1 ~+l '
where the k
+
l subscripts (not necessarily distinct) are taken from ~O,1,2, ... ,nJ,
is said to be 0/ type (k Il ) .Definition. The distance matrix D is
0/
type (k.t) if all vectors of type {k ,l}have nonpositive norm. For example. type (2.1) means
that is. the triangle inequality for g,iM;' E: C.
Lemma 2.2. Type (k ,k -1) implies type (k ,Ie).
frgQ!. From a vector of type (Ie ,k) we obtain 2k vectors of type (Ie ,Ie -1). hence 2k inequalities. Addition yields the required inequality, multiplied by the factor 2k -2.
Lemma 2.3. Type (Ie
+
2.1e+
1) implies type (k+
1.k). type (k+
1,k+
1) implies type (Ie ,Ie ) •.fI2Qf. Put.£i1
=
.!411A:+2·
-3-Definition. Dis hype'1"'lTl.etric if it is of type (k
+
l,k) for all k ::!!! 1. From the first lemma we infer [6]:Theorem 2.4. If D is hypermetric, then D is of negative type.
The converse is not true; in fact, the two notions are of a completely ditierent nature. Indeed, in the hyperplane
f:
xi=
0 any vector is near a rational vector,i=O
of which some integral multiple is of type (k ,k). The vectors of
f:
!t'i=
1 do not(=0
have this property. The first of the next theorems is simply a restatement of the definition.
Theorem 2.5. D is hypermetric iff all ;I
=
f:
!t'i.£.i with xi € Z,f;
:z;
=
1 have'=0 i=O
negative norm.
Theorem 2.6.
D
is of negative type iti all;I=
f;
%i.£.i withf:
xi.=
0 have negative(=0 i=O
norm, that is, iti D induces a negative definite inner product on the hyperplane
1:
!t'i=
O.i=O
l:r.QQ!. In
L
Xi = 0, vectors of positive norm are near rational vectors of positive norm, and would imply the existence of integral vectors of positive norm, quod non.3. the condition "negative type"
n
J--J---::::: ... 3
-4-Clearly. negative type means that the hyperplane
L:
%i = O. spinned by thedifferences of ,Q}.QI' ...•
.sn..
does not intersect the light cone. in other words. that the vector N perpendicular toL:
%l=
0 has positive norm. One wonderswhether this perhaps is automatically true. but it is not. as the following exam-ple due to P. Winkler [10] demonstrates. Consider the following matrix of size n
+
2:(Ji./i) 1 1 3!::.t
1 0 1 3!::.t
1 1 0 3!::.t
3!::. 3!::. u 2J-21
The submatrtx of size n + 1 obtained by deleting the first row and column is the Cayley-Menger matrix of the "corner"-simplex 1.2.3 .... ,n in
W-l
designed above; hence the submatrix has the required signature [2]: one positive and n negative eigenvalues. The whole matrix is the Gram matrix of these n+
1 vectors together with the vectorJi.
in RI.n which is defined byso that the whole matrix is singular. An easy calculation yields 2(N,/i)
=
5 - n, which is negative for n ~ 6. This provides examples in whichL:
~=
0 intersects the light cone: the corresponding distance matrix is not of negative type.In general. let
Ji.
e:
Rt.n be perpendicular to Ql - ,Q} • £2 -,Q}. . . . . Qn - £0. that isAbbreviate detGram to det. We have
% _ detUh - fa • .£2 - fa. . . . . .£n -fa}
1/2 - det{.Qo.Qt • .Q! •..•• Qn)
It follows that %
>
0 iff Gram (Ql -.Q) • . • . : Qn -fa} is negative definite {Schoenberg's condition [7]}. This Gram matrix is the difference of Gram(Ql',Qa .... . .£n) and a rank 1 matrix (2J in the example above, which again leads to indefiniteness for n ~ 6).The hypermetric condition is formulated in terms of lattices. It requires that the lattice induced in the fiat
L:
.:I;=
1 has no points inside the light cone. In other words. the intersection ofL:
.:I;=
1 and the light cone should be a hole in this
-5-We close this section with the following remark. It is well known [S.8] that the n pairs 1!Jo • .Ed. i
=
1. ...• n. provided with the distanceare isometrically embeddable in (n -1)-dimensional Euclidean space. This is stereograpbic projection from hyperbolic to Euclidean space.
4. Integral coordinates
Is it possible to view a distance matrix D of size n
+
1 as the Gram matrix of vectors with easy coordinates. such as lO.1~ or fO.1.-li?The following vectors in Rl.JA are good candidates: the
aJ:
tkJ
vectors !o..J"£ ± §il . . . ± ",. all having norm O. If D is an integral matrix with constant
row sums k then. according to an observation by M. Deza [5]. D can be represented by ~O.l ~-vectors as follows. Apart from the constant term Jilo..J"£ • each .E\ is given k coordinates one and elsewhere zero. Each pair .!::t and.Ej are taken such that they overlap in k -
dt,;
ones (apart from the Jilo-coordinate). Then each .!::t is isotropic. and the inner products C.!::t • .E;) equal the corresponding entries of D;i.i
e:
~O.l ... nJ.
If D has not constant rows, then by a theorem of Sinkhorn (9] there is a unique doubly stochastic matrix obtained from
D
by pre- and postmultiplication by a diagonal matrix. Thus, modulo diagonal matrices. the above construction still works for non-regular distance matrices. The problem remains to find the smallest dimension m+
1 for the space from which the representing vectors are taken.5.Ortbogonal.ity
The lattice ~ Xi S. ~
e:
Z, in Rl.n cannot be an orthogonal lattice since<st •
.E;) cannot vanish for independent vectors !Jo • .E!, ... • £n. On the other hand, the vectors .E1 -.Q). S! -~. . . . ,1m -.Q) can be orthogonal. namely wheneverthat is. whenever the distance matrix D has the entries
for
i.i
=
l, ...• n. apart from a row and a column (.ra • .QJ). An easy example has all C.!::t • .QJ)=
1. hence
-6-u
t
1
2J-21 .
6. The matrix D and its inverse
C -n
The diagonal of D-I consists of the numbers
n-
1 -- G ranl\J;lO.Ql.···'..Gn· '7." .. )D(gl.!:2 .... • Qn) D(Qo.!:2, ... ,Qn) D(Qo,Qt. ... ,Qn-l)
D(JiD.Ql • ... ,Qn-l.Qn)
D(Qo.Ql.!:2 • ... ,Qn) • D(Qo,Qlt!:2, ... • Qn)
The easiest example is the following:
I wonder whether §5 and §6 are of some value for the problems of metric embed-ding.
:References
[1]
P. Assouad. Y. Deza, &paces 'I'TUltriques plongeables dn:ns un hypercube. Annals DiscI'. Yath.Ji
(1980). 194-210.[2]
L.Y. Blumenthal. Theory and. applications0/
distance geometry, Oxford Univ. Press, 1953.[8] H.S.Y. Coxeter, Non-euclidean geometry, Toronto Univ. Press. 1942.
(4]
Y. Deza (= Y.E. Tylkin). On Hammi:ng geometry0/
unitary cubes, Doklady Akad. Nauk SSR ~ (1960). 1037-1040.
-7-[5] M. Deza, private communication.
[6] J.B. Kelly, Combinatorial inequalities, pp. 201-207 in "Combinatorial struc-tures and their applications", ed. R. Guy c.s., Gordon and Breach. 1970. [7] I.J. Schoenberg. Remarks to M. Freahet's article ... , Annals Math . .a§ (1935),
724-732.
[6] J.J. Seidel. Angles and. distances in n-d:imensiDnal Fluclid.ean and
ncm-FJu.clidean geometry, Proc. Kon. Ned. Akad. Wetensch. Q6 (= Indag. Math. J1) 329-340. 535-541 {1955}.
[9] R. Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices, Ann. Math. Statist. 35 (1964). 676-679.