On the dimension of some real, bounded rank, matrix spaces

ON THE DIMENSION OF SOME REAL, BOUNDED RANK, MA TRIX SP A CES Abstra ct. Given n ∈ N , let X b e either the set of hermitian or r e al n × n matric es of r ank at le ast n − 1 . If n i s even, we give a sharp estimate on the maximal dimension of a r e al ve ct or sp ac e V ⊂ X ∪ { 0 } . The rusults ar e ob- taine d, via K − the or y, by studying a bund le m ap induc e d by the adjugation of matric es. AMS ( MOS) Sub ject Cl assification: 15A30 (55N15, 19L64). Key words: Hermitian matric es, K − t he ory, adjugate matric es, subsp ac es of b ounde d r ank matric es. 1. Introduction: real subsp aces of ma trices Let n > 0 b e a p ositiv e in teger and denote by M n ( C ) the space of n × n complex matrices. Let H a nd R b e resp ectiv ely the real s u bspaces of M n ( C ) of hermitian and r eal matrices: H = { A ∈ M n ( C ) | A = t A } , R = { A ∈ M n ( C ) | A = A } , dim H = dim R = n 2 . Definition 1.1. F or any ve ctor subsp ac e V ∈ M n ( C ) , the minimal rank of V is the p ositive inte ger m V = min A ∈M n ( C ) \{ 0 } r ank A. Let n o w H m and R m b e resp ectiv ely the sets of linear subs p aces of H and R having minimal rank m . Definition 1.2. Fixe d inte gers n > 0 and 0 < m ≤ n , set h n,m = max V ∈H m dim V and r n,m = max V ∈R m dim V Note that h n,n ≤ h n,n − 1 ≤ · · · ≤ h n, 1 = n 2 and similarly for r n,m . Consider the follo wing: Problem Compute or give an estimate of h n,m and r n,m . F or n ∈ N , fact orise n = 2 a +4 b (2 k + 1) with a, b, k ∈ N , 0 ≤ a ≤ 3 and d efine the real and complex Radon-Hurwitz n umbers as ρ ( n ) = 2 a + 8 b , ρ C ( n ) = 2( a + 4 b ) + 2. In [1] and [2] Adams, Lax and Phillips sho w, by determining the maximal n umber of ev erywhere in dep endent v ector fields on a sph er e, that r n,n = ρ ( n ) and h n,n = ρ C ( n/ 2) + 1. In th is pap er we pro v e: 1 2 ON THE DIMENSION OF SOME REAL, BOUNDED RANK, MA TRIX SP ACES Theorem If n > 0 is an even i nte ger, then h n,n − 1 ≤ ρ C ( n ) and r n,n − 1 ≤ ρ C ( n ) and, if 8 divides n , then r n,n − 1 = ρ C ( n ) = ρ ( n ) . Our in terest in dealing with spaces of her m itian matrices of rank b ounded from b elo w primarly arised by studyin g the k ernel of the cup-p r o duct map Λ 2 H 1 ( X, C ) → H 2 ( X, C ) of a compact K ¨ ahler v ariet y X without Albanese fibrations (see th e au th or and Pirola in [7]). The problem p osed in this pap er is, how eve r, a particular formulati on of the follo win g more general question: giv en a set X of (real or complex) matrices v erifying some algebraic conditions such as e.g. symmetry or b ound edness of rank, what is the maximal dim en sion of a linear subsp ace V ⊂ X ∪ { 0 } ? This question has an interest on its o wn since it naturally arises in man y differen t settings. In its original formula tion (concerning real inv ertible ma- trices) it is equiv alen t to the p roblem of finding ev erywhere indep edent vecto r fields on a sphere and has b een solv ed, as said b efore, by Adams and others in [1, 2]. F or s ymmetric inv ertible matrices it is relat ed to sp ectral problems and PDE’s (see F riedland, Robbin and Sylve ster [11]). In the case of real rectangular matrices of maximal r ank it is related to the existence of b ilin- ear nonsingular maps R k × R n → R m generalizing the m ultiplication map of the classical division algebras ov er R (see Hu rwitz [12], Berger and F riedland [5], Lam and Yiu [14]). In th e case of constan t rank matrices it is related to the geometric dimension of v ector b undles ov er the pro jectiv e space, hence to immersion p oblems (see Adams [3], Beasley [4], the autor and Pirola [6], Landsb erg and Ilic [15 ], Mesh ulam [16], Rees [17], W est wick [20]). The pro of of our theorem is b ased on the follo wing considerations: if A ∈ M n ( C ) is a real or her m itian matrix of rank ≥ n − 1, th en its adjugate matrix A ∗ is not zero and ψ ( A ) = A + i t A ∗ is in v ertible. When n is eve n, ψ ( − A ) = − ψ ( A ) and, if V i s any real linear space of such matrices, the map ψ : V \ { 0 } → GL n ( C ) induces an isomormphism b et w een the trivial v ector bund le C n and n times the complex tautologica l bundle o v er P ( V ). This isomorphism, wh en it is read in the rin g of r educed K − theory of P ( V ), giv es an algebraic relatio n b et ween n and d im V whic h implies the stat ed estimate. The pap er is th erefore organized in t wo sections: in the fi rst one we recall the principal results ab ou t th e str ucture of the K − theory ring for the pro jectiv e space and we giv e a metho d f or relating linear spaces of matrices to bun dle maps. In the second part w e prov e ou r statemen ts ab out adju gate matrices and conclude the pro of of the th eorem. Ac kno wle dgements. Th e author w ould lik e to express h is gratitude to S. F riedland for the m an y fruitful discussions on these sub jects. The author is also deeply grateful to G. P . Pirola for the un coun table num b er of teac hings he ga v e and for his constan t interest in the author’s w ork. ON THE DIMENSION OF SOME REAL, BOUNDED RANK, MA TRIX SP ACES 3 2. Pre liminar y st a tements F or the m ain cited r esults ab out K-theory refer to [1] or to [13]; f or the definition and prop erties of o dd maps in this setting, the main references are [11] and [9]. 2.1. K-theory and Radon-Huwitz n umbers. Let RP d − 1 b e th e r eal pr o- jectiv e space of dimension d − 1, ξ its tautol ogical v ector bun d le and R k , C k the real and complex trivial bu ndles of ran k k . De note by ξ C = ξ ⊗ C the complexification of ξ and remark that ξ C = ξ ⊕ ξ and ξ ⊗ ξ = R . Recall that the ring of r ed uced complex K-theory e K C ( RP d − 1 ) is the ring of formal differences [ E ] − [ C k ] of (isomo rphism classes of ) complex ve ctor bund les ov er RP d − 1 suc h that E has rank k . Prop osition 2.1. The ring e K C ( RP d − 1 ) is isomorphic to the p olyno mial ring Z [ µ ] with the fol lowing r elations µ 2 = − 2 µ (1) µ g ( d )+1 = 0 , g ( d ) = inte ger p art of d − 1 2 . (2) The isomorph ism is g iven by the identific ation µ = [ ξ C ] − [ C ] Using the ab o v e identificat ion of rings, we ca n sh o w the follo wing prop ert y relating the r ing structure to the complex Radon-Hur witz n umbers: Prop osition 2.2. F or any p ositive inte ger n ∈ N , nµ = 0 in e K C ( RP d − 1 ) ⇐ ⇒ d ≤ ρ C ( n ) . Pr o of. First of all, remark th at relat ion 2 can b e equ iv alen tly written (using relation 1) as 2 g ( d ) µ = 0 so that nµ = 0 if and only if n is an inte ger m ultiple of 2 g ( d ) . W r ite no w n = 2 a (2 k + 1) with a, k ∈ N , then d ≤ ρ C ( n ) = 2 a + 2 ⇔ a ≥ d − 1 2 − 1 2 ⇔ a ≥ g ( d ) , where the last equiv alence is giv en by the fact that a is an int eger and g ( d ) is either ( d − 1) / 2 or ( d − 1) / 2 − 1 / 2 according to the parit y of d . In conclusion, nµ = 0 ⇔ 2 a (2 k + 1) µ = 0 ⇔ a ≥ g ( d ) ⇔ d ≤ ρ C ( n )  2.2. Odd maps. Definition 2.3. Any map ψ : S d − 1 → M n ( C ) verifying the r elation ψ ( − x ) = − ψ ( x ) is c al le d an o dd m ap . Prop osition 2.4. Any o dd map ψ induc es a morphism of ve ctor bund les over RP d − 1 : Ψ : C n → nξ C define d lo c al ly as Ψ([ x ] , v ) = ([ x ] , ψ ( x ) v ) . 4 ON THE DIMENSION OF SOME REAL, BOUNDED RANK, MA TRIX SP ACES Mor e over, if ψ ( x ) has r ank r for any x , then K = ke r Ψ and C = cok er Ψ ar e wel l- define d ve ctor bund les and the fol lowing i somorh ism holds: (3) K ⊕ nξ C = C n ⊕ C . Pr o of. Consider the m ap Ψ ′ : S d − 1 × C n → S d − 1 × C n Ψ ′ ( x, v ) = ( x, ψ ( x ) v ) . Since ψ is o dd, Ψ ′ is equiv ariant with resp ect to the actions of R ∗ on S d − 1 × C n giv en by f λ ( x, v ) =  λ | λ | x, v  , g λ ( x, v ) =  λ | λ | x, λv  , λ ∈ R ∗ . The map indu ced b y Ψ ′ b y p assing to the qu otien ts is exactly Ψ , ind eed: S d − 1 × C n f λ = R P d − 1 × C n and S d − 1 × C n g λ = n ( ξ ⊕ ξ ) = n ξ C . Isomorphism 3 is a consequen ce of the fact th at an y exact sequence of vect or bund les sp lits.  With this setting, we can pro v e the main theorem of Adams, Lax and Phillips in [2] Theorem 2.5. If V is a r e al ve ctor sp ac e such that V \ { 0 } ⊂ G L n ( C ) , then dim V ≤ ρ C ( n ) . Pr o of. Apply p rop osition 2.4 to the follo wing setting: d = dim V , S d − 1 the unit sph ere of V and ψ the indu ced inclusion (wic h trivially is an o d d map) S d − 1 ⊂ V \ { 0 } ⊂ GL n ( C ) ⊂ M n ( C ) . In this case ψ ( x ) is alw a ys an inv ertible matrix, then isomorphism 3 b ecomes nξ C = C n . In the rin g e K C ( RP d − 1 ) this means th at n times the generator µ is zero, hence, by prop osition 2.2, d ≤ ρ C ( n ) .  3. Ad juga te ma t rices F or a square matrix A ∈ M n ( C ), let A i,j b e the submatrix obtained from A b y deleting its i -t h ro w and j -th column and denote by A c the transp ose of the adjugate of A : ( A c ) i,j = ( − 1) i + j det A i,j . W e define a map ψ : M n ( C ) → M n ( C ) b y setting ψ ( A ) = A + i A c , the bar denoting complex conju gation. It is clear fr om the definition of A c that if rank A ≤ n − 2, th en ψ ( A ) = A . No w set: Z = { A ∈ M n ( C ) | rank A ≥ n − 1 , det A 6 = ir, r < 0 } and remark th at b oth the sp aces H and R (of herm itian and r eal n × n m atrices) are subsets of Z . ON THE DIMENSION OF SOME REAL, BOUNDED RANK, MA TRIX SP ACES 5 Prop osition 3.1. F or ψ as ab ove, ψ ( Z ) ⊂ GL n ( C ) Pr o of. By con tradiction, tak e A ∈ Z and v ∈ C n , v 6 = 0 s uc h that (4) ψ ( A ) v = Av + i A c v = 0 . Multiplying this equ ation on the left by t A giv es (5) t AAv + i det( A ) v = 0 Then, since v 6 = 0, equation 5 sa ys that − i det( A ) is an eigen v ector of the hermitian m atrix t AA and is a fortiori real and not negativ e. This means that det( A ) = ir with r ≤ 0 and, since A ∈ Z , the equalit y det( A ) = 0 = det( A ) holds. Equation 5 b ecomes t AAv = 0 and, after left multi plication by t v , t v t AAv = k Av k = 0 . As a consequence, equation 4 reads (6) Av = A c v = 0 . F rom the fu n damen tal prop ert y of the adjugate matrix A t A c = d et( A ) I = 0 it follo ws that the image of t A c is co nt ained in the n u ll sp ace of A whic h is, b y equation 6 , spanned by v . Then, A c v = 0 implies A c t A c = 0 , that is A c = 0 bu t this is im p ossible since r ank A ≥ n − 1 forces at least one elemen t of A c to b e d ifferent from zero.  Remark 3.2. Set X = { A ∈ M n ( C ) | r ank A ≤ n − 2 } . It is somehow inter esting to observe that the family of maps ψ s : M n ( C ) → M n ( C ) ψ s ( A ) = A + s i A c , s ∈ R r e alizes, for 0 ≤ s ≤ 1 a homoto py fr om the identity id ( A ) = A to ψ = ψ 1 , fixing the algebr aic variety X p ointwise. Mor e over, for any s > 0 , ψ s ( Z ) ⊂ GL n ( C ) . Multilinearit y of the determinant imples that: Prop osition 3.3. If n is even, then ψ ( − A ) = − ψ ( A ) . Pr o of. It is a direct consequence of the fact that the submatrices A i,j ha v e o dd order hence det( − A i,j ) = − det( A i,j ) and consequently ( − A ) c = − A c .  W e can no w pro v e: Prop osition 3.4. If V is a r e al ve ctor sp ac e such that V \ { 0 } ⊂ Z and n is even, then dim V ≤ ρ C ( n ) . 6 ON THE DIMENSION OF SOME REAL, BOUNDED RANK, MA TRIX SP ACES Pr o of. The restriction of ψ to the u nit sphere of V is an o d d map ψ : S d − 1 → GL n ( C ) d = dim V ; b y prop osition 2.4, ψ indu ces an isomorphism of vect or bun dles o ver RP d − 1 : C n = n ξ C . In the r ing of r educed complex K-theory e K C ( RP d − 1 ) this isomorphim means that n times the generato r µ is zero, hence prop osition 2.2 implies d ≤ ρ C ( n ) .  As a corollary it follo ws: Theorem 3.5. If n is even, then h n,n − 1 ≤ ρ C ( n ) and r n,n − 1 ≤ ρ C ( n ) . Pr o of. This is due to the r emark that b oth H and R are subsets of Z .  3.1. Some remarks on the estimates. Th eorem 3.5 p ro vides a v ery sharp estimate on th e n umbers h n,n − 1 and r n,n − 1 . This is particularly clear when w e compare our result with the v alues of h n,n and r n,n giv en in [2]. Indeed, for h er m itian matrices, if we write as customary n = 2 a (2 k + 1 ) with a, k ∈ N , w e get h n,n = ρ C ( n/ 2) + 1 = 2 a + 1 ≤ h n,n − 1 ≤ 2 a + 2 = ρ C ( n ) ( n eve n) . F or real matrices, on the other h and, w e similarly ha v e r n,n = ρ ( n ) ≤ r n,n − 1 ≤ ρ C ( n ) ( n ev en) . It is in teresting to compare the v alues of ρ ( n ) to th ose of ρ C ( n ): writing n = 2 a +4 b (2 k + 1), with a, b, k ∈ N and 0 ≤ a ≤ 3, w e get the follo w ing a ρ ( n ) ρ C ( n ) 0 1+8b 2+8b 1 2 + 8b 4 +8b 2 4+8b 6+8b 3 8+8b 8+8b th us, in particular, Prop osition 3.6. If 8 divides n then r n,n − 1 = ρ ( n ) . This p rop osition giv es a n egativ e answer to a question p osed in [9] asking whether the inequ alities in the sequen ce r n,n ≤ r n,n − 1 ≤ · · · ≤ r n, 2 ≤ r n, 1 w ere alw a ys sharp or not. ON THE DIMENSION OF SOME REAL, BOUNDED RANK, MA TRIX SP ACES 7 Referen ces [1] J.F. Adams. V e ctor fields on spher es. A nn. of Math. (2) 75 ( 1962) 60 3-632. [2] J.F. Adams, P . Lax and R. Phillips. On matric es whose r e al l ine ar c ombinations ar e non-singular. Proc. Amer. Math. So c. 16 (1965) 318-322. [3] J.F. A dams. Ge ometric dim ension of bun d l es over RP n . Pro c. Int. Conf on Prosp ects of Math., Kyoto (1973) 1-17. [4] L. B. Beasley . 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