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Some Multilinear Algebra

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Some Multilinear Algebra Some multilinear algebra Bernhard Leeb January 25, 2020 Contents 1 Multilinear maps 1 1.1 General multilinear maps . 1 1.2 The sign of a permutation . 2 1.3 Symmetric multilinear maps . 3 1.4 Alternatingmultilinearmaps............................... 4 1.4.1 The determinant of a matrix . 6 1.4.2 The determinant of an endomorphism . 7 1.4.3 Orientation . 8 1.4.4 Determinant and volume . 10 2 Tensors 12 2.1 The tensor product of vector spaces . 12 2.2 The tensor algebra of a vector space . 17 2.3 The exterior algebra of a vector space . 21 See also: Lang Algebra,vanderWaerdenAlgebra I. 1 Multilinear maps 1.1 General multilinear maps We work with vector spaces over a fixed field K. Let V1,...,Vk,W be vector spaces. A map µ V1 ... Vk W × × → 1 is called multilinear if it is linear in each variable, that is, if all maps µ v1,...,vi 1, ,vi 1,...,vk Vi W are linear. The multilinear maps form a vector space Mult V1,...,Vk−; W . + ( ⋅ ) ∶ Multilinear maps in k variables are also called k-linear.The1-linearmapsarethelinear → ( ) maps, the 2-linear maps are the bilinear maps. Multilinear maps are determined by their values on bases, and these values are independent of each other. More precisely, if e i j J are bases of the V ,thenamultilinearmapµ is ji i i i determined by the values µ e 1 ,...,e( ) k W for j ,...,j J ... J ,andthesevalues j1 jk 1 k 1 k ( ) ( ( ) ∈ ) can be arbitrary, i.e. for any vectors wj1...jk W there is a unique multilinear map µ with µ e 1 ,...,e k w .Indeed,forthevaluesongeneralvectors( ) ∈ ( ) ∈ × v × a e i V ,we j1 jk j1...jk i ji Ji iji ji i ( ) ( ) ∈ ( ) obtain the representation ∈ ( ) = = ∑ ∈ k µ v ,...,v a µ e 1 ,...,e k . (1.1) 1 k iji j1 jk j1,...,jk i 1 ( ) ( ) In particular, if the dimensions( of the) = vector spaces= are⋅ ( finite, then ) dim Mult V1,...,Vk; W dim Vj dim W. j ( ) = ⋅ Examples: Products in algebras. Composition Hom U, V Hom V,W Hom U, W .Scalarprodutcs,symplecticforms, volume forms, determinant. Natural pairing V V K.( ) × ( ) → ( ) ∗ × → 1.2 The sign of a permutation Let Sk denote the symmetric group of k symbols, realized as the group of bijective self-maps of the set 1,...,k .Thesign of a permutation ⇡ Sk is defined as ⇡ i ⇡ j { } sgn ⇡ ∈ 1 . 1 i j k i j ( ) − ( ) ( ) = ∈ {± } It counts the parity of the number of inversions≤ < ≤ of− ⇡, i.e. of pairs i, j such that i j and ⇡ i ⇡ j .Thesignispositiveifandonlyifthenumberofinversionsiseven,andsuch permutations are called even. Transpositions are odd, and a permutation( ) is even if and< only if it( can) > be( written) as the product of an even number of transpositions. The sign is multiplicative, sgn ⇡⇡ sgn ⇡ sgn ⇡ , ′ ′ i.e. the sign map ( ) = ( ) ( ) sgn Sk 1 is a homomorphism of groups. Its kernel An →is called{± } the alternating group. Briefly, the sign map is characterized as the unique homomorphism Sk 1 which maps transpositions to 1. The remarkable fact is that such a homomorphism exists at all. A consequence is that, when representing a permutation as a product of transpositions,→ {± } the parity of the number of factors− is well-defined. Remark. For n 5thealternatinggroupAn is non-abelian and simple. ≥ 2 1.3 Symmetric multilinear maps We now consider multilinear maps whose variables take their values in the same vector space V .Onecallssuchmapsalsomultilinearmapson V (instead of on V n). We abbreviate Multk V ; W Mult V,...,V ; W k ( ) ∶= ( ) and denote by Multk V Multk V ; K the space of k-linear forms on V .ThenMult1 V V and, by convention, Mult0 V K. ∗ ( ) ∶= ( ) ( ) = A k-linear map ( ) = µ V k V ... V W k = × × → is called symmetric if µ vσ 1 ,...,vσ k µ v1,...,vk (1.2) ( ) ( ) for all permutations σ Sk and all( vi V . ) = ( ) Remark. There is a natural∈ action ∈ Sk Multk V ; W (1.3) of permutations on multilinear forms given by ( ) σµ v1,...,vk µ vσ 1 ,...,vσ k . ( ) ( ) Indeed, ( )( ) = ( ) ⌧ σµ v1,... σµ v⌧ 1 ,... µ v⌧ σ 1 ,... ⌧ µ v1,... ( ) ( ( )) for σ,⌧ Sk.Thuscondition(1.2)canberewrittenas( ( ))( ) = ( )( ) = ( ) = (( ) )( ) ∈ σµ µ, i.e. µ is symmetric if and only if it is a fixed point= for the natural Sk-action (1.3). We describe the data necessary to determine a symmetric multilinear map. If ei i I is a basis of V ,thenµ Multk V is symmetric if and only if the values on basis vectors satisfy µ ei ,...,ei µ ei1 ,...,eik for all i1,...,ik I and σ Sk,compare ( ∈ ) σ 1 σ k ∈ ( ) the representation (1.1) of( ) the values( ) on general vectors. Hence, if I is equipped with a total ( ) = ( ) ∈ ∈ ordering “ ”, then µ is deteremined by the values µ ei1 ,...,eik for i1 ... ik, and these values can be arbitrary. ( ) If dimensions are finite, we conclude that dim V k 1 dim Multsym V ; W dim W. k k + − ( ) = ⋅ Further discussion: Polynomials and symmetric multilinear forms. 3 1.4 Alternating multilinear maps We now consider a modified symmetry condition for multilinear maps, namely which is “twisted” by the signum homomorphism on permutations: Definition (Antisymmetric). Amapµ Multk V ; W is called anti- or skew-symmetric if µ vσ 1 ,...,vσ k ∈ sgn(σ µ )v1,...,vk (1.4) ( ) ( ) for all vi V and permutations( σ Sk. ) = ( ) ⋅ ( ) In terms∈ of the natural action∈Sk Multk V ; W we can rewrite (1.4) as σµ sgn( σ )µ, (1.5) for all σ Sk. = ( ) ⋅ A closely related family of conditions will turn out to be more natural to work with: ∈ Lemma 1.6. The following three conditions on a k-linear map µ Multk V ; W are equivalent: (i) µ v ,...,v 0wheneverv v for some 1 i k. 1 k i i 1 ∈ ( ) (ii) µ v ,...,v 0wheneverv v+ for some 1 i j k. ( 1 k) = i= j ≤ < (iii) µ v ,...,v 0wheneverthev are linearly dependent. ( 1 k) = = i ≤ < ≤ They imply that µ is antisymmetric. ( ) = Proof. Obviously (iii) (ii) (i). (i) antisymmetric and (ii): Suppose that (i) holds. The computation ⇒ ⇒ ⇒ β u, v β v,u β u v,u v β u, u β v,v for bilinear maps β shows( that) + then( (1.4)) = holds( + in the+ general) − ( k-linear) − ( case) for the transpositions of pairs i, i 1 of adjacent numbers. Since these transpositions generate the group Sk,itfollows 1 that (1.4) holds for all permutations σ Sk, that is, µ is antisymmetric. The antisymmetry together( with+ ) (i) implies (ii). ∈ (ii) (iii): Suppose that the vi are linearly dependent. In view of the antisymmetry (implied already by (i), as we just saw), we may assume that vk is a linear combination of the other vi, ⇒ that is, vk i k aivi.Thenµ v1,...,vk i k aiµ v1,...,vk 1,vi 0becauseof(ii). < < − Definition= ∑ (Alternating). Amap( µ ) =Mult∑ k V ;(W is called alternating) = if it satisfies (one of) the equivalent conditions (i-iii) of the lemma. ∈ ( ) According to the lemma, alternating multilinear maps are antisymmetric. If char K 2, then also the converse holds:2 1 The permutations≠ σ Sk, for which (1.4) holds, form a subgroup of Sk. 2The field K has characteristic 2, if 2 1 1 0inK. In this case, 2 has a multiplicative inverse in K, i.e. one can divide by 2 in∈ K. On the other hand, if the field K has characteristic 2, i.e. if 2 0inK,then 1 1 and hence a a for all a K≠, i.e. there∶= + are ≠no signs in K. = − = − = ∈ 4 Lemma. If char K 2, then antisymmetric multilinear maps are alternating. Proof. It suffices to≠ treat the bilinear case. An antisymmetric bilinear map β satisfies β v,v β v,v for all v V ,andhence2β v,v 0. Dividing by 2 yields that β is alternating. ( ) = − (Moreover,) in characteristic∈ 2antisymmetryandsymmetryare“transverse”conditions;( ) = the only multilinear maps, which are both symmetric and skew-symmetric, are the null-maps. ≠ Remark. If char K 2, then bilinear forms can be uniquely decomposed as sums of symmetric and alternating ones, since ≠ β u, v β v,u β u, v β v,u β u, v . 2 2 ( ) + ( ) ( ) + ( ) ( ) = symmetric + anti-symmetric If char K 2, then the relations between the conditions are di↵erent. Since there are no signs, skew-symmetry is the same as symmetry, whereas alternation is more restrictive if k 2. = Since antisymmetry coincides with either alternation or symmetry, depending on the char- ≥ acteristic, alternation is the more interesting condition to consider. We denote by Altk V ; W Multk V ; W the K-vectorspace of alternating multilinear( ) maps,⊂ and( we write) Altk V Altk V ; K . We now describe the data determining an alternating multilinear map.( ) ∶= ( ) Lemma 1.7. If ei i I is a basis of V ,then↵ Multk V ; W is alternating if and only if (i) ↵ e ,...,e 0ifsomeofthee agree, and i1 (ik ∈ ) ij ∈ ( ) (ii) ↵ e ,...,e sgn σ ↵ e ,...,e for all σ S if the e are pairwise di↵erent. ( iσ 1 )i=σ k i1 ik k ij ( ) ( ) Proof. The( conditions are) = obviously( ) ⋅ necessary.( ) ∈ To see that they are also sufficient, we first treat the case k 2ofabilinearformβ.Fora vector v i viei,assumptions(i+ii)imply = 2 = ∑ β v,v vi β ei,ei vivj β ei,ej β ej,ei 0, i i j 0 0 ( ) = ( ) + ( ( ) + ( )) = where we assume that I is equipped= with a total ordering “= ”. Thus, β is alternating. In the general k-linear case, it follows that the bilinear forms ↵ e ,...,e , , ,e ,...,e i1 ij 1 ij 2 ik for 1 j k are alternating, and consequently all bilinear forms ↵ v1,...,vj− 1, , ,vj+ 2,...,vk ( ⋅ ⋅ ) since they are linear combinations of the former.
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