Seidel, J.J.

English

Pure OAI Repository

 Approximation numbersCitation for published version (APA):Seidel, J. J. (1979). Approximation numbers: extended notes of a lecture by A.A. Melkman. (EindhovenUniversity of Technology : Dept of Mathematics : memorandum; Vol. 7910). Technische Hogeschool Eindhoven.Document status and date:Published: 01/01/1979Document 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 beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI 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 pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand 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, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 17. Nov. 2023EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics Memorandum 1979 - 10 September 1979 Approximation Numbers Extended Notes of a Lecture by A.A. Melkman by J. J. Seidel Technological University Department of Mathematics P.O. Box 513, Eindhoven The Netherlands Approximation numbers. Extended notes of a lecture by A.A. Melkman, THE, August 1979. by J.J. Seidel 1. Summary. The problem is to approximate, in the p-norm, the unit matrix I of size m m by rank n matrices, n $ m, that is, to determine the numbers a n = a (I ; Q,m n m 1 := min A e: A n (I m where A denotes the set of all m x m matrices of rank $ n. n Since only the extremes of the octahedron I I xii $ 1 are of importance this is equivalent to a n = min A e: A n max l$kSm i=l where e 1, ••• ,e denote the columns of I. In terms of subspaces x c :!Rm - -m m n this is equivalent to a n = min X c lRm n min x e: x n It follows from a theorem by Sofman [5] that a (I;!lm !lm)=Yl-n n m 1 ' 2 m Melkman [ 1 ] proves that for p ;i. 2 - ~11 p a (I n m i; ) ;i. i_ 1 + (m _ 1 ) [ <.m - 1) (m - n) ] n 1 p 1- 1 + -2 (p - 1) .. p For the case p = co this amounts to a (I n m - 2 -( \-1 + I (m - 1) n ~ \ 1 ) \ m - n Moreover, Melkman shows that for p ~ 2 equality holds if and only if there exists a regular two-graph on m vertices with the multiplicities n and m - n. 2. Eutactic stars ([4] ,[3]) Let x denote a real inner product space with an orthonormal basis m e e Let P · X + X be a projection operator (linear, symmetric, -1'"""'-m" · m m idempotent) , and let PX = X Then, for k , !l = 1 , .•. , m, m n hence for the vectors Pe1, ... ,Pe EX the Gram matrix equals the coordinate - -m n matrix with respect to ~ 1 , ... ~m. This matrix is symmetric and idempotent, has trace n hence m I (P~k' P~k) = n • k=1 By definition, the vectors Pe1, ••. ,Pe constitute a eutactic star. This - -m star is spherical whenever all (P~k,P~k) are equal. If, in addition, \ (P~k' P~.Q.) \ is constant for all k ~ .Q., then the lines spanned by P~1 , ••• ,P~m constitute a set of m equiangular lines I m - n in X at cos Cj} = ( 1 ) . Such a n nm -set is extremal in the following sense. For any set of m equiangular lines in JRn at angle 1jJ, let G denote the Gram matrix of a set of m unit vectors, one along each of the lines. Then G = I + c cos ijJ, where c is a symmetric matrix of size m with diagonal zero and entries + 1 elsewhere. Since G is positive semidefinite of rank n, its nonzero eigenvalues A1, ••• ,An satisfy m = tr G A 1 + • . . + An , m + m ( m - 1 ) cos 2 1jJ = tr G 2 = A~ + • . . + A~ m2 2 m - n 2 hence -- ~ m + m(m - 1) cos 1jJ ~ cos 1jJ n ' n (m - 1) - 3 -Equality holds iff Al = = A , that is, iff C has just two eigenvalues, n of multiplicities n and m - n. A triple of equiangular lines is of acute or obtuse type, according as the lines are spanned by a triple of equiangular vectors at acute or at ob-tuse angle. An extremal set of m equiangular lines in lRn is characterized by the property that each pair of lines is in a constant number of triples of obtuse type. This is equivalent to the existence of a regular two-graph on n vertices, whose eigenvalues have the multiplicities n and m - n. For the definition and a survey cf. [3] . 3. Sofman's theorem ([5], see also [1]). Theorem. Let ~ 1 , .•• ,~m be an orthonormal basis of xm. m The conditions I ~~ = n, 0 ~ ~k k=l ~ 1 are necessary and sufficient the existence of a subspace for X of X such that, fork= 1, ••. ,m, n m the projection of ~k onto Xn has length ~k. Proof. The necessity of the condition has been observed in 2. For the suffi -ciency we use induction on n. Suppose, fork = 1, .•• ,m. m 2 Then I ~k k=2 0 ~ ~k ~ 1, max k 2 = n - ~ z n - 1, 1 o ~ n 2 ~ ~ 2 ' ••• ' o ~ nm ~ ~m' ~2 + ~2 + + ~2 = 1 2 · • · m n • so we may choose n 1 ~ .• ,nm such that n1 m 2 I nk = n - 1 . k=l 0, By the induction hypothesis there exists Xn-l c Xm such that, for k = 1, ... ,m, the projection of ~k onto X 1 has length nk . Then X 1 ~ e . We define n- n- -1 X := <e > ffi X . Then, fork =1, ••. ,m,the proJ·ections z of e_k onto X n -1 n-1 -k n have lengths n2 , ··.,Sm = n and m, m 2 I i;;k k=l n • - 4 -~1 = ~1 ---------------------------------x m x n e~ -2 ~2 I we wish to rotate ~1 , . •• ,~m so as to move step by step from r,; 1 , ••• ~m to s , ..• ,s . Take any k = 2, ..• ,m, and consider a rotation about a in the 1 m plane < ~l '~k > , leaving all other ~i fixed: ~l (a.) = ~l cos a.- ~k sin a., ~k(a.) = ~ 1 sin a. + ~k cos a, e. (a) -]. Then r;j (a) := II P(~j (a.)) II satisfy r;~ (a) + r;~ (a.) 2 2 = r,; 1 + ~ , and r;. (a.) = r; . for i I= 1,k ]. l. Since r;k(a) is an increasing function of a from 1 , and since 1 choose a. such that z;k (a.) = sk. Then z; . (a.) ]. = z;i s; si implies m 2 m s~ l i;i (a.) = n = l and r;i s; s1 s; i; 1 ( CJ. ) . i=l i=l l. Thus we have made z;k into ~ , and z; 1 (a.) is again the largest number . Now e . -l. repeat the process with each of the indices I= 1, so as to arrive at an or-thonormal basis whose projections onto X have given lengths s 2 , ..• ,~ n m. Then also the first length fits with ~. This finishes the induction step from n - 1 to n. For n = 1 the theorem is true, since then the line spanned by the vector (s1 , ... ,~) applies. Thus the proof is completed. The theorem may be rephrased in the following ways. Corollary. Necessary and sufficient for the existence of a eutactic star in xn consisting of m vectors at lengths s11· · ·1 ~m are s~ + ..• + s~ = n, 0 ~ sk ~ 1 • x n-1 - 5 -Corollary. Necessary and sufficient for the existence of a symmetric idem-potent matrix C are: trace C = rank C diag C ;;i. 0 In particular,taking ~l = ~ = t; = V - we have m m Corollary. Corollary. Corollary; X contains spherical eutactic stars of any cardinality m. n Given m, n e: JN, m 2: n, there exists a symmetric zero-diagonal matrix of size m whose only eigenvalues are n and n - m. a (I n m g,m 1 Proof. For tjle t 2-norm we know min ::k - ~ II = ::k - P;:k II x e: x n Now II P:: l 112 + ... + Pe 112 = n , -m II ::1 - Pe 11 2 + ... + II e - Pe 11 2 = m - n -1 -m -m implies max II ::k - P ( ::k ) II ;;i. 1 :::; k :5m j 1 - n m By Sofman' s theorem, an Xn with II P:: 111 = = II Pe II = n really exists, -m m hence for such X n max 1 :::; k:5 m Remark. Any X , for which n max II ::k - Pe 1 :::; k:::; -k m satisfies II P=111 = II II = I Pe 11 -m 1 - n m 1 n - -m j !!. . Indeed, m 0 - 6 -II p~k 11 2 + II =k - P=k 11 2 = 1 I m 2 2 n + I 11 e - P~k II = m n+m max ll =k - P=k II I k=l -k 1 s;k::;; m II 2 11 e - 2 Vl:Sk;S;m 11 e - Pe = max P~k II I -k -k 1 s;. k s; -k m hence all II P=k II are equal. 4. The theorem of Hahn - Banach. Let v be a Banach space with norm II II , and let W be a closed subspace. The quotient space V/W is a Banach space with norm 11 v + w II : = inf II~ - :!. II • w €. w * The dual space V of all continuous linear functionals f on V is a Banach space with norm II f II* = sup I f (!_) I 11 x 11 = 1 x €. v = sup 0 ,.~E V I f<~l I llxll The following is a consequence of the Hahn - Banach theorem. Theorem. 11~11 sup II f 11* = *1 f €. v I f <xl I Furthermore, we need the following If <xl I = sup 0 ,. fE v* II f 11* Theorem. There exists an isometric isomorphism of the spaces (V/w) * and w~ : = {f "' v* I f (W) = O} • We apply the above to the linear space X provided with the 2 -norm m p II x II - p - 7 -Holder with 1 1 -+-p p' 1, 1 < p < co • * As a consequence, x has t ,-norm m p 1 · crl jy.1p')P' 1 llyll p' = + l. / p For a subspace X of X we now have n m 1 = 1 ' y E p' .l * xn = {y E xm I y 1x 1 + ••• + y x = O, V } mm ~EXn * x m where y, = y(e.). Apply Hahn-Banach to the quotient V/W = X /X, then we i -i m n obtain (since the dimension is finite) Theorem . l! ~k + Xn lip = inf X E X n II ~k - ~ ll P = In 6. we will use the following consequence of Holder's inequality 1 1 Theorem • r Indeed, apply Holder with q;;;. 1 to the (m - 1) -vectors . , jx jrandl,. m 1- .!. Ix. I r ~ (m - 1) q ( I J j;ifk 1 jx . lrq)q , J .~. ___! - .!. 1 (m - 1) qr r ~ ( \' I I qr\ qr \j~k xj ) For p = r, that is q = 1, our inequality is an equality . , for p ;;;. r . . 1 , then For p > r, equality holds iff the (m - 1)-vectors are proportional . - 8 -5. Melkman's theorem for p = 00 • In the following proof we will use back and forth the consequence of Hahn-Banach exposed in 4. In addition, we will use the Cauchy-Schwarz inequality in the following form: m l i=l i;ifk IY. 1. 1 <. j I J. i=l i;ifk \y. 12 V m - 1 • J. We first prove -the p = ~ case, since this case is representative. Theorem. a (I n m '},m 1 1 . i m :;;;., ( 1 +/<m-lJn \-~ ~ m-n } ' and equality holds iff there exists a regular two-graph on m ver-tices with multiplicities n, m - n . Proof. Fork c {1,2, ... ,m} we have :;;;., max IYk l = Orfy .L X - 1 j m Jyil2- 1Yk! 2 - - n IYk I + I/ m I i=l l1 JI = !( ~) -2 = max + V m - 1 1 0-f:y.L X - - n L ll y ll2 i- j ( max I Yk I f 2 1-1 = l! + I/ m - 1 lj Q. f .z E x~ II z 112 J - 9 -t + j r \-2 1-1 = v iU - 1 min II e - x "2; 1 J \ -k x e: x n Since f (z) := ~ + V m - 1 J z-2 - 1 ] -\s a monotone increasing function of z, this implies min max min II e - x II ~ -k - 00 x c x 1 :s;k ::;;m x e: x n m n ~ ~ + - 1 j ( min vm max min \x c x l~k~m x e: n m Thus we have expressed m m a (I ; t1 , 200) in terms of n m 1 -1 \-2 II e - ~" 2; -1J -k x n which equals J 1 - !!. by 3. Substitution yields the inequality of the theorem. m Now suppose we have equality a (I n m 2m 2m) = '1 + j n(m - l) J -l . 1 , co Ll m - n We analyse the various steps performed in the proof of the inequality. First, for an optimal X we have n max l ~k~m x ~n x II ::k - ~I 2 n n m hence, by the remark at the end of 3, min II ::k - xii / 1 n for all k = - - , -2 m x e: x n In addition, for any 1 ~ k ~ m, the x which achieves 1, ... ,m . min II ::k - ~II oo , also minimizes II ::k - ~lb . Hence the matrix A which approximates I in the t -norm m oo also works for the t2-norm. - 10 -secondly, if equality holds in Cauchy-Schwarz, then the corresponding vectors of length m - 1 and (1,1, ... , \/ , ... ,1) are proportional. So we may take IY 1 1= ... = IYml = 1 except for IYkl which equals / (m - 1) (m - n) n since 1 I . n(m - 1) 1 + m - n For each k = 1, ••. ,m we find such a vector y_, which is proportional to the projection of e onto X~ . These vectors are taken as the columns of the -k n following coordinate matrix B ( = Gram matrix, cf. 2) of rank m - n : I :2, e:12 "1~ y e:2m I e: . . I 1 ' e: . . = e: .. I . l.J· l.J J l. B = l ~ml IYkl =I (m-1) (m-n) y= n e:in2 y J It follows that the lines spanned by these vectors are equiangular, at cos2x = (m _ l)7m _ n) . This set of equiangular lines in X~ is extremal in the sense of 2 (just interchange n and m - n). Equivalently, X contains an extremal n 2 m - n set of equiangular lines at cos ~ = ~(-m~-~1-)-n- . Equivalently, there exists a regular two-graph on m vertices with the multiplicities n and m - n. 6. Melkman's theorem for p > 2 . For p ~ r, a lower bound for a (I n m m ; z1 m m m Z ) in terms of a (I ; i1 ,tr) is p n m obtained, and specialized to the case r = 2, since a (I n m n = = / 1 m is known. Instead of Cauchy-Schwarz we use the following conse-quence of Holder, cf.4 : - 11 -1 r' for p > r ; for p = r this degenerates into an equality. For a fixed k, we abbreviate as follows: n p : = min X E X n Then for p > r , p' ( n ) p = max x. ..L x n ( r''f' l I YJ. I rjrfk max 1 _ E_'._ -( I Yk I 1 + (m - 1) r' --llyll I r p'~ 1 + (m - 1) 1 - ~ ( max y ..L x -- n It follows that Theorem . n > p For p > 2 a (I ,Q,m n m 1 ~ 1 p' p' ( ( ) -r' - 1) ;t. nr , J - £.'.._ (m - 1) 1 r' I -p ,Q,; ) > 11 + (m - l)cm - !) (m - n))2(p-1 ) 1 -1 + -p . .. - 12 -and equality holds iff there exists a regular two-graph on m vertices with multiplicities n, m - n. Proof. Put r = 2 in the formula for 1\i' proceed as in 5 , and substitute the value of a for r = 2. This yields the inequality. For the case of n equality we must have equality in the consequence of Holder's inequality. Since p > 2, again the(m - 1)-vectors are proportional, and the reasoning of 5 works. Remark. The second part of the reasoning above does not work for p = 2. Indeed, then the consequance of H6lder's inequality is an equality, and yields nothing new. In fact, for p = 2 we do have a (I n m im 2 n m but the extremal sets need not be extremal sets of equiangular lines . Any spherical eutactic star provides an extremal set. As an example we mention the root systems [41. [1] A.A. Melkman, The distance of a subspace of Rm from its axes and n-widths of octahedra, to be published. [2] A. Pinkus, Matrices and n-widths, to appear, Lin. Alg.and Appl . [3] J.J. Seidel, A survey of two-graphs, Pree.Intern. Coll.Teorie Combinatorie (Roma, 1973), Accad. Naz. Lincei , Roma 1976, Vol I, 481 - 511. [4] J.J. Seidel, Eutactic stars, Coll. Math. Soc.J. Bolyai 18, Combina-torics , Keszthely 1976, 983 - 999. [5] L. B. Sofman, Diameters of octahedra, Mat. Notes 5 (1969) 258 - 262. 

Approximation numbers:extended notes of a lecture by A.A. Melkman

Seidel, JJ

Repository TU/e

 Approximation numbersCitation for published version (APA):Seidel, J. J. (1979). Approximation numbers: extended notes of a lecture by A.A. Melkman. (EindhovenUniversity of Technology : Dept of Mathematics : memorandum; Vol. 7910). Technische Hogeschool Eindhoven.Document status and date:Published: 01/01/1979Document 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 beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI 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 pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand 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, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 16. Nov. 2023EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics Memorandum 1979 - 10 September 1979 Approximation Numbers Extended Notes of a Lecture by A.A. Melkman by J. J. Seidel Technological University Department of Mathematics P.O. Box 513, Eindhoven The Netherlands Approximation numbers. Extended notes of a lecture by A.A. Melkman, THE, August 1979. by J.J. Seidel 1. Summary. The problem is to approximate, in the p-norm, the unit matrix I of size m m by rank n matrices, n $ m, that is, to determine the numbers a n = a (I ; Q,m n m 1 := min A e: A n (I m where A denotes the set of all m x m matrices of rank $ n. n Since only the extremes of the octahedron I I xii $ 1 are of importance this is equivalent to a n = min A e: A n max l$kSm i=l where e 1, ••• ,e denote the columns of I. In terms of subspaces x c :!Rm - -m m n this is equivalent to a n = min X c lRm n min x e: x n It follows from a theorem by Sofman [5] that a (I;!lm !lm)=Yl-n n m 1 ' 2 m Melkman [ 1 ] proves that for p ;i. 2 - ~11 p a (I n m i; ) ;i. i_ 1 + (m _ 1 ) [ <.m - 1) (m - n) ] n 1 p 1- 1 + -2 (p - 1) .. p For the case p = co this amounts to a (I n m - 2 -( \-1 + I (m - 1) n ~ \ 1 ) \ m - n Moreover, Melkman shows that for p ~ 2 equality holds if and only if there exists a regular two-graph on m vertices with the multiplicities n and m - n. 2. Eutactic stars ([4] ,[3]) Let x denote a real inner product space with an orthonormal basis m e e Let P · X + X be a projection operator (linear, symmetric, -1'"""'-m" · m m idempotent) , and let PX = X Then, for k , !l = 1 , .•. , m, m n hence for the vectors Pe1, ... ,Pe EX the Gram matrix equals the coordinate - -m n matrix with respect to ~ 1 , ... ~m. This matrix is symmetric and idempotent, has trace n hence m I (P~k' P~k) = n • k=1 By definition, the vectors Pe1, ••. ,Pe constitute a eutactic star. This - -m star is spherical whenever all (P~k,P~k) are equal. If, in addition, \ (P~k' P~.Q.) \ is constant for all k ~ .Q., then the lines spanned by P~1 , ••• ,P~m constitute a set of m equiangular lines I m - n in X at cos Cj} = ( 1 ) . Such a n nm -set is extremal in the following sense. For any set of m equiangular lines in JRn at angle 1jJ, let G denote the Gram matrix of a set of m unit vectors, one along each of the lines. Then G = I + c cos ijJ, where c is a symmetric matrix of size m with diagonal zero and entries + 1 elsewhere. Since G is positive semidefinite of rank n, its nonzero eigenvalues A1, ••• ,An satisfy m = tr G A 1 + • . . + An , m + m ( m - 1 ) cos 2 1jJ = tr G 2 = A~ + • . . + A~ m2 2 m - n 2 hence -- ~ m + m(m - 1) cos 1jJ ~ cos 1jJ n ' n (m - 1) - 3 -Equality holds iff Al = = A , that is, iff C has just two eigenvalues, n of multiplicities n and m - n. A triple of equiangular lines is of acute or obtuse type, according as the lines are spanned by a triple of equiangular vectors at acute or at ob-tuse angle. An extremal set of m equiangular lines in lRn is characterized by the property that each pair of lines is in a constant number of triples of obtuse type. This is equivalent to the existence of a regular two-graph on n vertices, whose eigenvalues have the multiplicities n and m - n. For the definition and a survey cf. [3] . 3. Sofman's theorem ([5], see also [1]). Theorem. Let ~ 1 , .•• ,~m be an orthonormal basis of xm. m The conditions I ~~ = n, 0 ~ ~k k=l ~ 1 are necessary and sufficient the existence of a subspace for X of X such that, fork= 1, ••. ,m, n m the projection of ~k onto Xn has length ~k. Proof. The necessity of the condition has been observed in 2. For the suffi -ciency we use induction on n. Suppose, fork = 1, .•• ,m. m 2 Then I ~k k=2 0 ~ ~k ~ 1, max k 2 = n - ~ z n - 1, 1 o ~ n 2 ~ ~ 2 ' ••• ' o ~ nm ~ ~m' ~2 + ~2 + + ~2 = 1 2 · • · m n • so we may choose n 1 ~ .• ,nm such that n1 m 2 I nk = n - 1 . k=l 0, By the induction hypothesis there exists Xn-l c Xm such that, for k = 1, ... ,m, the projection of ~k onto X 1 has length nk . Then X 1 ~ e . We define n- n- -1 X := <e > ffi X . Then, fork =1, ••. ,m,the proJ·ections z of e_k onto X n -1 n-1 -k n have lengths n2 , ··.,Sm = n and m, m 2 I i;;k k=l n • - 4 -~1 = ~1 ---------------------------------x m x n e~ -2 ~2 I we wish to rotate ~1 , . •• ,~m so as to move step by step from r,; 1 , ••• ~m to s , ..• ,s . Take any k = 2, ..• ,m, and consider a rotation about a in the 1 m plane < ~l '~k > , leaving all other ~i fixed: ~l (a.) = ~l cos a.- ~k sin a., ~k(a.) = ~ 1 sin a. + ~k cos a, e. (a) -]. Then r;j (a) := II P(~j (a.)) II satisfy r;~ (a) + r;~ (a.) 2 2 = r,; 1 + ~ , and r;. (a.) = r; . for i I= 1,k ]. l. Since r;k(a) is an increasing function of a from 1 , and since 1 choose a. such that z;k (a.) = sk. Then z; . (a.) ]. = z;i s; si implies m 2 m s~ l i;i (a.) = n = l and r;i s; s1 s; i; 1 ( CJ. ) . i=l i=l l. Thus we have made z;k into ~ , and z; 1 (a.) is again the largest number . Now e . -l. repeat the process with each of the indices I= 1, so as to arrive at an or-thonormal basis whose projections onto X have given lengths s 2 , ..• ,~ n m. Then also the first length fits with ~. This finishes the induction step from n - 1 to n. For n = 1 the theorem is true, since then the line spanned by the vector (s1 , ... ,~) applies. Thus the proof is completed. The theorem may be rephrased in the following ways. Corollary. Necessary and sufficient for the existence of a eutactic star in xn consisting of m vectors at lengths s11· · ·1 ~m are s~ + ..• + s~ = n, 0 ~ sk ~ 1 • x n-1 - 5 -Corollary. Necessary and sufficient for the existence of a symmetric idem-potent matrix C are: trace C = rank C diag C ;;i. 0 In particular,taking ~l = ~ = t; = V - we have m m Corollary. Corollary. Corollary; X contains spherical eutactic stars of any cardinality m. n Given m, n e: JN, m 2: n, there exists a symmetric zero-diagonal matrix of size m whose only eigenvalues are n and n - m. a (I n m g,m 1 Proof. For tjle t 2-norm we know min ::k - ~ II = ::k - P;:k II x e: x n Now II P:: l 112 + ... + Pe 112 = n , -m II ::1 - Pe 11 2 + ... + II e - Pe 11 2 = m - n -1 -m -m implies max II ::k - P ( ::k ) II ;;i. 1 :::; k :5m j 1 - n m By Sofman' s theorem, an Xn with II P:: 111 = = II Pe II = n really exists, -m m hence for such X n max 1 :::; k:5 m Remark. Any X , for which n max II ::k - Pe 1 :::; k:::; -k m satisfies II P=111 = II II = I Pe 11 -m 1 - n m 1 n - -m j !!. . Indeed, m 0 - 6 -II p~k 11 2 + II =k - P=k 11 2 = 1 I m 2 2 n + I 11 e - P~k II = m n+m max ll =k - P=k II I k=l -k 1 s;k::;; m II 2 11 e - 2 Vl:Sk;S;m 11 e - Pe = max P~k II I -k -k 1 s;. k s; -k m hence all II P=k II are equal. 4. The theorem of Hahn - Banach. Let v be a Banach space with norm II II , and let W be a closed subspace. The quotient space V/W is a Banach space with norm 11 v + w II : = inf II~ - :!. II • w €. w * The dual space V of all continuous linear functionals f on V is a Banach space with norm II f II* = sup I f (!_) I 11 x 11 = 1 x €. v = sup 0 ,.~E V I f<~l I llxll The following is a consequence of the Hahn - Banach theorem. Theorem. 11~11 sup II f 11* = *1 f €. v I f <xl I Furthermore, we need the following If <xl I = sup 0 ,. fE v* II f 11* Theorem. There exists an isometric isomorphism of the spaces (V/w) * and w~ : = {f "' v* I f (W) = O} • We apply the above to the linear space X provided with the 2 -norm m p II x II - p - 7 -Holder with 1 1 -+-p p' 1, 1 < p < co • * As a consequence, x has t ,-norm m p 1 · crl jy.1p')P' 1 llyll p' = + l. / p For a subspace X of X we now have n m 1 = 1 ' y E p' .l * xn = {y E xm I y 1x 1 + ••• + y x = O, V } mm ~EXn * x m where y, = y(e.). Apply Hahn-Banach to the quotient V/W = X /X, then we i -i m n obtain (since the dimension is finite) Theorem . l! ~k + Xn lip = inf X E X n II ~k - ~ ll P = In 6. we will use the following consequence of Holder's inequality 1 1 Theorem • r Indeed, apply Holder with q;;;. 1 to the (m - 1) -vectors . , jx jrandl,. m 1- .!. Ix. I r ~ (m - 1) q ( I J j;ifk 1 jx . lrq)q , J .~. ___! - .!. 1 (m - 1) qr r ~ ( \' I I qr\ qr \j~k xj ) For p = r, that is q = 1, our inequality is an equality . , for p ;;;. r . . 1 , then For p > r, equality holds iff the (m - 1)-vectors are proportional . - 8 -5. Melkman's theorem for p = 00 • In the following proof we will use back and forth the consequence of Hahn-Banach exposed in 4. In addition, we will use the Cauchy-Schwarz inequality in the following form: m l i=l i;ifk IY. 1. 1 <. j I J. i=l i;ifk \y. 12 V m - 1 • J. We first prove -the p = ~ case, since this case is representative. Theorem. a (I n m '},m 1 1 . i m :;;;., ( 1 +/<m-lJn \-~ ~ m-n } ' and equality holds iff there exists a regular two-graph on m ver-tices with multiplicities n, m - n . Proof. Fork c {1,2, ... ,m} we have :;;;., max IYk l = Orfy .L X - 1 j m Jyil2- 1Yk! 2 - - n IYk I + I/ m I i=l l1 JI = !( ~) -2 = max + V m - 1 1 0-f:y.L X - - n L ll y ll2 i- j ( max I Yk I f 2 1-1 = l! + I/ m - 1 lj Q. f .z E x~ II z 112 J - 9 -t + j r \-2 1-1 = v iU - 1 min II e - x "2; 1 J \ -k x e: x n Since f (z) := ~ + V m - 1 J z-2 - 1 ] -\s a monotone increasing function of z, this implies min max min II e - x II ~ -k - 00 x c x 1 :s;k ::;;m x e: x n m n ~ ~ + - 1 j ( min vm max min \x c x l~k~m x e: n m Thus we have expressed m m a (I ; t1 , 200) in terms of n m 1 -1 \-2 II e - ~" 2; -1J -k x n which equals J 1 - !!. by 3. Substitution yields the inequality of the theorem. m Now suppose we have equality a (I n m 2m 2m) = '1 + j n(m - l) J -l . 1 , co Ll m - n We analyse the various steps performed in the proof of the inequality. First, for an optimal X we have n max l ~k~m x ~n x II ::k - ~I 2 n n m hence, by the remark at the end of 3, min II ::k - xii / 1 n for all k = - - , -2 m x e: x n In addition, for any 1 ~ k ~ m, the x which achieves 1, ... ,m . min II ::k - ~II oo , also minimizes II ::k - ~lb . Hence the matrix A which approximates I in the t -norm m oo also works for the t2-norm. - 10 -secondly, if equality holds in Cauchy-Schwarz, then the corresponding vectors of length m - 1 and (1,1, ... , \/ , ... ,1) are proportional. So we may take IY 1 1= ... = IYml = 1 except for IYkl which equals / (m - 1) (m - n) n since 1 I . n(m - 1) 1 + m - n For each k = 1, ••. ,m we find such a vector y_, which is proportional to the projection of e onto X~ . These vectors are taken as the columns of the -k n following coordinate matrix B ( = Gram matrix, cf. 2) of rank m - n : I :2, e:12 "1~ y e:2m I e: . . I 1 ' e: . . = e: .. I . l.J· l.J J l. B = l ~ml IYkl =I (m-1) (m-n) y= n e:in2 y J It follows that the lines spanned by these vectors are equiangular, at cos2x = (m _ l)7m _ n) . This set of equiangular lines in X~ is extremal in the sense of 2 (just interchange n and m - n). Equivalently, X contains an extremal n 2 m - n set of equiangular lines at cos ~ = ~(-m~-~1-)-n- . Equivalently, there exists a regular two-graph on m vertices with the multiplicities n and m - n. 6. Melkman's theorem for p > 2 . For p ~ r, a lower bound for a (I n m m ; z1 m m m Z ) in terms of a (I ; i1 ,tr) is p n m obtained, and specialized to the case r = 2, since a (I n m n = = / 1 m is known. Instead of Cauchy-Schwarz we use the following conse-quence of Holder, cf.4 : - 11 -1 r' for p > r ; for p = r this degenerates into an equality. For a fixed k, we abbreviate as follows: n p : = min X E X n Then for p > r , p' ( n ) p = max x. ..L x n ( r''f' l I YJ. I rjrfk max 1 _ E_'._ -( I Yk I 1 + (m - 1) r' --llyll I r p'~ 1 + (m - 1) 1 - ~ ( max y ..L x -- n It follows that Theorem . n > p For p > 2 a (I ,Q,m n m 1 ~ 1 p' p' ( ( ) -r' - 1) ;t. nr , J - £.'.._ (m - 1) 1 r' I -p ,Q,; ) > 11 + (m - l)cm - !) (m - n))2(p-1 ) 1 -1 + -p . .. - 12 -and equality holds iff there exists a regular two-graph on m vertices with multiplicities n, m - n. Proof. Put r = 2 in the formula for 1\i' proceed as in 5 , and substitute the value of a for r = 2. This yields the inequality. For the case of n equality we must have equality in the consequence of Holder's inequality. Since p > 2, again the(m - 1)-vectors are proportional, and the reasoning of 5 works. Remark. The second part of the reasoning above does not work for p = 2. Indeed, then the consequance of H6lder's inequality is an equality, and yields nothing new. In fact, for p = 2 we do have a (I n m im 2 n m but the extremal sets need not be extremal sets of equiangular lines . Any spherical eutactic star provides an extremal set. As an example we mention the root systems [41. [1] A.A. Melkman, The distance of a subspace of Rm from its axes and n-widths of octahedra, to be published. [2] A. Pinkus, Matrices and n-widths, to appear, Lin. Alg.and Appl . [3] J.J. Seidel, A survey of two-graphs, Pree.Intern. Coll.Teorie Combinatorie (Roma, 1973), Accad. Naz. Lincei , Roma 1976, Vol I, 481 - 511. [4] J.J. Seidel, Eutactic stars, Coll. Math. Soc.J. Bolyai 18, Combina-torics , Keszthely 1976, 983 - 999. [5] L. B. Sofman, Diameters of octahedra, Mat. Notes 5 (1969) 258 - 262. 

NARCIS 