Unit vectors with non-negative inner products

Transcription

1 Unit vectors with non-negative inner proucts Bos, A.; Seiel, J.J. Publishe: 01/01/1980 Document Version Publisher s PDF, also known as Version of Recor (inclues final page, issue an volume numbers) Please check the ocument version of this publication: A submitte manuscript is the author's version of the article upon submission an before peer-review. There can be important ifferences between the submitte version an the official publishe version of recor. People intereste in the research are avise to contact the author for the final version of the publication, or visit the DOI to the publisher's website. The final author version an the galley proof are versions of the publication after peer review. The final publishe version features the final layout of the paper incluing the volume, issue an page numbers. Link to publication Citation for publishe version (APA): Bos, A., & Seiel, J. J. (1980). Unit vectors with non-negative inner proucts. (Einhoven University of Technology : Dept of Mathematics : memoranum; Vol. 8010). Einhoven: Technische Hogeschool Einhoven. General rights Copyright an moral rights for the publications mae accessible in the public portal are retaine by the authors an/or other copyright owners an it is a conition of accessing publications that users recognise an abie by the legal requirements associate with these rights. Users may ownloa an print one copy of any publication from the public portal for the purpose of private stuy or research. You may not further istribute the material or use it for any profit-making activity or commercial gain You may freely istribute the URL ientifying the publication in the public portal? Take own policy If you believe that this ocument breaches copyright please contact us proviing etails, an we will remove access to the work immeiately an investigate your claim. Downloa ate: 08. Oct. 018

2 TECHNOLOGICAL UNIVERSITY EINDHOVEN Department of Mathematics The Netherlans Memoranum May 1980 Unit vectors with non-negative inner proucts by A. Bas an J.J. Seiel

3 - 1 - Unit vectors with non-negative inner proucts A. Bos an J.J. Seiel 1. Introuction In [lj W. Kruskal is creite with the following Conjecture The square length of the sum of n unit vectors in E, having mutually non-negative inner proucts, is at least n + r(-r) _ ' where n = r (mo ), 0 ~ r <. Moreover, equality is attaine if an only if the n vectors are sprea as evenly as possible over an orthonormal set of vectors. For a number of cases we settle this conjecture in the affirmative. Moreover, we escribe a setting for the problem which may lea to a general proof. However, the general conjecture remains open.. The problem (cf. [lj). Suppose we have + 1 observations of n stanarize variables. Arrange them in an (+1) x n matrix X= [x.. J; i lj 1,..., + 1 j 1,...,n, an assume that they are nonnegatively correlate, that is, for j,k = 1,...,n assume where 1 +1 r J. k : = 1 L ( x.. - x. ) (x. k - x k ) ~ 0, + i=l lj J l x. J L i=l +1 _ x..,!l L (x.. - x J ' ) lj i=l lj 1. n The sum variable y. = L x.. achieves its maximum possible variance l lj j=l n if all correlations r equal 1. It is natural to ientify the jk

4 - - II rel ateaness" of the variables -wteh the--variance of t heir sum an ask what is the minimum possible variance. Without changing the situation essentially we may assume that +1 the column sums of X are 0, L x.. = a for all j. If x. enotes the j-th column, then the variance i=1 1.J -J of x. equals 1 1 (x.,x. ), which is 1 by assumption. Also the correlation -J :J -J IT r jk = +1 (~j'~) ~ 0, hence no angle between the xj's excees /. The variance we wish to minimize can be written as 1 (n n) +1 L~., L~ j=1 J j=1 J 1 Now write u. := x. These vectors are all perpenicular to -J I+1-J (1,1,.,1)t. Hence we have n unit vectors u,,u in 1 ~ with non- - -n n negative inner proucts,,an the problem -is to minimize " J=1.L ~ II J. 3. Inequalities Let S enote the co~lection of all sets of n unit vectors in ~ n, all of whose inner proucts are nonnegative. Let n = q + r, 0 $ r <. For any S E Sn,' let G = [gij] enote the Gram matrix of S, an let IT = A1 ~ A~ ~A enote the nonzero eigenvalues of G. Lemma 3.1. n + ( lt-n) (-l) Proof. tr G = nan tr G = Lg~. 1.J rea n - IT, IT, whence (n - IT)

5 - 3 - implying the first inequality. The secon follows from 0 S gij S 1, an the thir one is implie by choosing x = (1,1,,1) in 1T?: (Gx,x)/(x,x). o When oes equality hol? In the first inequality iff A= ~A that is iff (= A say), 1T(1T - A)P, where P = [p,p,] is the rank one matrix mae up from the (positive) 1. J components p, of the unit eigenvector of 1T. G is a (0,1) matrix iff 1. equality hols in the secon, an G has constant row sums iff equality hols in the thir inequality. Finally, our inequalities imply n S 1T, an equality hols if an only if G = I I n /, that is, iff S consists of orthonormal vectors each repeate n/ tim~s. Part of the conjecture reas Conjecture 3.. Ig"?: (n + r(-r»/, S E S. 1.J n, Clearly, lemma 3.1 implies that conjecture 3. is true for n = 0 (mo ). We observe that the right-han sie of the inequality equals the sum of the entries of I J 1 r q+ [ o the ajacency matrix of Turan's graph, cf. []. This illustrates the following lemma, by which conjecture 3. nees only be investigate for irreucible S E S. n, Lemma 3.3. If n = n 1 + n, = 1 +, n = q + r, n 1 = ql l + r 1, n =q + r, 0 S r <, 0 S r 1 < 1, 0 S r <, then - r ) 1 + n + re - r) o.

6 - 4 - Proof. Suppose ql ~ q' then ql ~ q ~ q. Inee, hence ql ~ n < (q + l). Straightforwar caluculation shows that the left-han sie of the inequality in the lemma equals 1 ( (q - ql) + (q - ql)) - r 1 (q - ql) + ( (q - q) - (q - q)) + r (q - q) Since r 1 < 1 an r ~ 0, this is not less than 1 «q - ql) - (q - ql)) + «q - q) - (q - q)), which is nonnegative, since q - ql an q - q are nonnegative integers. Remark. In the lemma equality hols iff q q = q + 1, r = o. Remark. If conjecture 3. were true, then the Perron eigenvalue TI G woul satisfy of n r (-r) n -+ ~ 7T. 4. The solution in a special case Theorem 4.1. The conjecture is true for S E S ' S a two-istance n, -1 set with inner proucts 0 an a -1 Proof. Let G = I + a A with a (0,1) matrix A having m ones. Thus, - a is the smallest eigenvalue of A. Assume the conjecture were not

7 true for any irreucible I + a A. Lemma 3.1 an the assumption then yiel n 0 n + m a n + m o < n + r(-r) FromQ)an@we obtain o n(n - ) $ m < a(n - r) (n + r - ). For n ~ the right han inequality yiels a contraiction. For < n ~ we have a nr $ m < or < 4r < n, since 0 < ~ <, hence m < n +r an A is the ajacency matrix of a tree. But n - 1 = m < r < n is impossible. We are left with n >, but then o < (n-r) (n-+r) n (n-) 1 + r(-r) n (n-) ~ s In [3J it is prove that any graph of iameter D has smallest eigenvalue -0 $ - cos 'IT / (D + ). Hence our graph has iameter D 1, a = 1, 1, r 0, contraicting ~. This proves the theorem. Corollary. The ajacency matrix of a graph has Perron-eigenvalue ~ 0 (n - + r) (n - r) / n, where (-0), of multiplicity n - n = q + r, 0 $ r <., is the smallest eigenvalue an

8 Geometric methos Let S-l -- {x E JR I } (x,x) = 1 The hyperplane perpenicular to any unit vector Z E JR etermines two close hemispheres {x E S-l I (x,z) ~ O} H { -l I ( ) } an = XES X,Z ~ 0. -l For any finite set XES the convex hull C(X) is the set of all finite convex linear combinations of elements of X, that is, C(X) := {z E s-l I Z Its ual spherical polytope D(X) is efine by D(X) := {z E S-l I 'v' XEX (X,Z) ~ a}, that is, the intersection of the positive hemispheres of the vectors of X. Let P an P * be spherical polytopes. P * is sai to be' ual to P if ~ : F(P) ~ F(P * ) is a bijection from the set of faces of P to the set of faces of P * such that f ~ g ~ ~f ~ ~g for all f,g E F(P). P is calle self-ual if p* = P. The polar set P of a spherical polytope P is efine by -l I P := {z E S V XEP (X,Z) ~ a}. Clearly P is ual to P an D(X) = C(x) Theorem 5.1. Assume that X is such that for all YES. Then X E V(DX», where V(P) is the set of vertices of P. n,

9 - 7 - Proof. Suppose x E X is not a vertex of D(X), that is, there exist a,b E D(X) such that x = aa + Sb, where 0 < a, S < 1 are relate by a + as (a,b) + S = 1. Now let X' = X \ {xl Then ( L z,x) ZEX' a( L z,a) + S( L Z,b) ZEX' ZEX' is, as we shall prove, a nonconstant concave function of a, thus reaching its mininum for a = 0, say. But if X" = X' u {a}, this contraicts the assumption, since then because ( L z, L Z) = (I Z, ZEX ZEX ZEX' + ( L z,a) + 1 ZEX' L ~) + ( L Z'X} + 1 > (L Z, ZEX ZEX" ZEX' ( L z, L z). ZEX" ZEX" f(a) is a concave function iff f ~ O. Hence the sum of two concave a functions an the square root of a nonnegative concave function are again concave. Since -a (a,b) (a,b) -+ 1 ' a it remains to prove that a {(a,b) - this is obvious since (a,b) ~ 1. I} is a concave function of a, an Corollary 5..: For every x E X, with X as in theorem 5.1, there exist - 1 linearly inepenent xi E X such that (x,x ) i = 0 for all i 1,,, - 1.

10 - 8 - Corollary 5.3. For = an Ixi n == 1 (mo ), Z X " L Z" Equality is attaine iff X consists of two orthogonal vectors, each repea t e --- n-l an --- n+l. t1mes respect1ve. 1 y. Corollary 5.4. If X contains a set of orthonormal vectors, then D(X) = C(X) is the regular orthogonal spherical polytope spanne by these vectors an n + r(-r) Theorem 5.5. Assume that X is as in thm. 5.1, then a self-ual spherical polytope P exists with X ~ V(P) Proof: From the properties of X we know C(X) ~ D(X) an further V(C(X» = X ~ V(D(X». Let L be the set of all polytopes P with P P an V(C(X» ~ V(P) ~ V(D(X». Clearly C(X) E L, so L is not empty. The set L is partially orere by pi < P iff V(P) C V(P'). L contains an upper boun of each totally orere subset M of L, so, with the lemma of Zorn, L contains a maximal element, which has to be self-ual an which contains X = V(C(X» as vertices. 6. Cases in which the conjecture hols The conjecture hols for i) n ~, all. Equality hols iff all vectors are orthonormal; ii) n == 0 (mo ), all. See the observations after conjecture 3..; iii) =, all n. Corollary 5.3.; iv) all n, all, some special cases as in section 4 an corollary 5.4.

11 - 9 - References [1J J.B. Kruskal & H.J. Witsenhausen, An inequality for positively correlate variables, Journal Amer. Statistical Ass., 69 (1974), [J P. Turan, On the Theory of graphs, Collog. Math. (1954), [3J R. Zurmuhl, Matrizen, Ch. 17.5, Springer 1958.