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do not edit, text has been copied out 2.8 The Outer Product of Tensors and Use of Going back to our economical Picture A notation, consider two vectors which transform in the usual rank-1 tensor manner relative to some underlying transformation F (for which R is the linearization), a'i = ΣjRijaj b'k = ΣmRkmbm from (2.1.5) (2.8.1) where we temporarily show the summation symbols. Multiplying these equations together gives (a'i)(b'k) =(ΣjRijaj)(ΣmRkmbm) = Σjm RijRkm(ajbm) and then hiding the sums again, (a'ib'k) = RijRkm(ajbm) . (2.8.2) But looking at (2.1.6), we see that this object is transforming as a rank-2 tensor, therefore it is a rank-2 tensor, and we can write it as M'ik = RijRkmMjm where Mij ≡ aibj . (2.8.3) The rank-2 tensor Mij = aibj is said to be the outer product of two rank-1 tensors (vectors). This idea can be generalized ad infinitum. For example, if K is a rank-2 tensor and v is a vector, then Mijk = Kijvk (2.8.4) is a rank-3 tensor because it transforms as one, using the same argument shown above. Consider, Mabcde = KabKcd ve . (2.8.5) If K is a rank-2 tensor and v is a rank-1 tensor, then M is a rank-5 tensor. Of course since this is a "true tensor equation", indices may be shuffled any way one wants, such as Mabcde = KabKcd ve . (2.8.6) Just imagine applying g** several times to both sides of (2.8.5) to get (2.8.6). There are so many possibilities for creating outer product tensors that one sometimes forgets that not all tensors can be "factored" into products of lower rank tensors. The Symbol Appears In Sections 1.1 and 1.2 we had vw being an element of a "tensor product space" VW and we described two approaches to the development of the meaning of the symbol : quotient space and category theory. Here we provide a third approach to the meaning of which is equivalent to that of the first two approaches. This third approach is geared to dealing with tensor components so there are lots of indices flying around, whereas in Sections 1.1 and 1.2 components were not even mentioned. Recall our previous two equations Mabcde = KabKcd ve . (2.8.5) Mabcde = KabKcd ve . (2.8.6) In order to display the outer product as a unified entity, we had to make up a new symbol M to represent the outer product tensor. We can avoid having to do this by writing M = KKv, so that the symbol in our "third approach" is just a way to name an outer product tensor. The above equations are then (KKv)abcde = KabKcd ve . (2.8.7) (KKv)abcde = KabKcd ve . (2.8.8) Here K and v are tensors, they are not spaces, so this is more like vw than VW. In fact, as a special case we can use this idea to name the outer product of two vectors to be rank-2 tensor ab, (ab)ij = aibj a,b ϵ V . (2.8.9) Notice that is a non-commuting operator: ab ≠ ba . In our "third approach" the symbol exists only within the context of VV, since vectors a and b both belong to the x-space of Picture A which we identify with vector space V. However, one can extend this meaning of to apply more generally as the outer product of vectors in different vector spaces, (vw)ij = viwj v ϵ V w ϵ W . (2.8.10) If we try to fit this into our notion of tensor transformations, we would need two copies of Picture A, one for U→V and the other for X→W with vector transformations v(V)i = R(V)ijv(U)j R(V)ij = linearization of some transformation x' = F(V)(x) w(W)i = R(W)ijw(X)j . R(W)ij = linearization of some transformation y' = F(W)(y) (2.8.11) Then the transformation of the outer product "tensor" would be written as, (v(V)iw(W)i) = R(V)iaR(W)jb (v(U)aw(X)b) or [(vw)(V,W)] ij = R(V)iaR(W)jb [(vw)(U,X)]ab UX → VW (2.8.12) One might refer to (vw)(V,W) as a "cross space rank-2 tensor". Normally the word "tensor" is used when W = V. Then the above reads, [(vw)(V,V)] ij = R(V)iaR(V)jb [(vw)(U,U)]ab UU → VV or [(vw)(e)] ij = RiaRjb [(vw)(u)]ab // Picture E (2.6.1) or (vw)' ij = RiaRjb (vw)ab // Picture A (2.1.1) (2.8.13) As for bases, one can consider (ab)ij = aibj of (2.8.9) in several situations depending on the spaces in which the two vectors lie. Here are two examples ( the basis must of course match on the two sides ) [(ab)(e)]ij = a(e)ib(e)j [(ab)(e,u)]ij = a(e)ib(u)j . (2.8.14) As an example of the second equation, one could write [(enem)(e,u)]ij = (en(e))i (em(u))j = δni Rmj (2.8.15) where the last expression comes from (2.6.4). As a final outer product example, consider the outer product of three vectors, Mijk = aibjck a,b,c ϵ V (2.8.16) Using our naming method for outer products, this becomes (abc)ijk = aibjck (2.8.17) with this obvious extension to the outer product of any number of vectors (abc....)ijk.... = aibjck.... (2.8.18) 2.9 The Inner Product (Contraction) of Tensors It is easy to show that, due to the orthogonality rules (2.1.8), internal index contractions within a tensor structure behave as a scalar, which is to say, behave as if they weren't there at all. Such contractions in a tensor structure reduce the rank of the tensor by two, resulting in an "inner product". The contracting sum must occur on a "tilted pair" of indices. The standard first example to consider is this (as usual, sums on repeated indices), Mij = aibj = a rank-2 tensor, which we now contract to form: s = Mii = aibi = a rank-0 tensor (a scalar) (2.9.1) Using our notation (2.2.5) this is written s = a b (2.9.2) which is an "inner product" of two vectors. This is of course the inner product / scalar product / dot product which makes our vector space V be a Hilbert space. In this example, creating an "inner product" of the two vectors ai and bj which has rank-0 is going in the opposite direction of the "outer product" that creates Mij = Mij = aibj of rank-2. The term "contraction" is more often applied to reducing the rank of tensors than is "inner product", and perhaps it is best to reserve the term "inner product" for the above dot product of two vectors. A few other examples of rank reduction by contraction. Define Mabcd ≡ KabQcd = rank-4 tensor (2.9.3) Tac ≡ Mabcb = KabQcb = rank-2 tensor (2.9.4) In this last example, contraction on the b index happens to occur between the two rank-2 tensors from which M was constructed as an outer product. One more step, S ≡ Taa = KabQab = rank-0 tensor (scalar) (2.9.5) Using the notation introduced in the previous section, we can write the inner product a b as a contraction of the outer product ab s = (ab)ii = (ab)ii and ||a||2 ≡ aiai = (aa)ii (2.9.6) Similarly (2.9.3,4,5) can be written (KQ)abcd = KabQcd = rank-4 tensor (2.9.7) Tac = (KQ)abcb = rank-2 tensor (2.9.8) S = Taa = (KQ)abab = rank-0 tensor (scalar) (2.9.9) Recall that (2.2.5) defines the dot product of two vectors in V a b = gijaibj = gijaibj = aibi = aibi x-space = V (2.2.5) It is possible to define an inner product operator for use between two elements of VV : (ab) (cd) ≡ Σij(ab)ij(cd)ij (ab), (cd) ϵ VV (2.9.10) = Σijaibjcidj . With this definition, we have a tiny theorem: Theorem: (ab) (cd) = (ac)(bd) a,b,c,d ϵ V (2.9.11) Proof: (ac)(bd) = (Σi aici)( Σj bjdj) = Σij aicibjdj = Σij aibj cidj = Σij(ab)ij(cd)ij = (ab) (cd) . Suppose dim(V) = n and dim(W) = n'. Then we can extend the above theorem to VW in this way. First define the dot product as, (ab') (cd') ≡ Σi=1nΣj=1n'(ab')ij(cd')ij (vw),(v'w') ϵ VW (2.9.12) = Σijaib'jcid'j The corresponding Theorem is then Theorem: (ab') (cd') = (a c)(b' d') a,c ϵ V b',d' ϵ W (2.9.13) Proof: (ac)(b'd') = (Σi=1n aici)( Σj=1n b'jd'j) = Σijaicib'jd'j = Σijaib'jcid'j = Σij (ab')ij (cd')ij = (ab') (cd') In a similar fashion one can show using (2.8.15) that with the following definition, (abc) (def) ≡ Σijk (abc)ijk (def)ijk VVV (2.9.14) one obtains Theorem: (abc) (def) = (a d)(b e)(c f) all vectors ϵ V (2.9.15) with a similar extension to VWX, Theorem: (ab'c") (de'f") = (a d)(b' e')(c" f") a,d ϵ V; b',e' ϵ W; c",f"' ϵ X (2.9.16) 2.10 Tensor Expansions Having a name for the outer product of two vectors allows us to write expansions of tensors of rank greater than 1 in a compact notation. Recall from ** that for rank-1 we had the expansion V = Σa [V(e)]a ea (2.10.1) For a rank-2 tensor M we then write (Picture E) M = Σab [M(e)]ab eaeb . (2.10.2) Let's try to make this be an identity by taking a tensor component of both sides in the e-basis [M(e)]ij = { Σab [M(e)]ab eaeb}(e)ij = Σab [M(e)]ab (eaeb)(e)ij = Σab [M(e)]ab (ea(e))i (eb(e))j // (2.8.9) = Σab [M(e)]ab δaiδbj // (2.6.4) [ col 3 row 3] = [M(e)]ij (2.10.3) so things are consistent. A convenient notational method for projecting out the coefficients of any tensor expansion is by using the tensor-product-space dot products defined in Section 2.9. To demonstrate, we us a tensor expansion in VW where the basis vectors are en and e'n for V and W, M = Σab [M(e,e')]ab eae'b M ϵ VW (2.10.4) The appropriate projector is (eie'j), which is just the expansion's basis eae'b with up/down toggled on the indices, and dummy variables like i,j selected. Using this projector one finds, (eie'j) M = Σab [M(e,e')]ab (eie'j) (eae'b) = Σab [M(e,e')]ab (ei ea)(e'j e'b) // theorem (2.9.13) = Σab [M(e,e')]ab δia δjb // dual pairs as in (2.3.2) = [M(e,e')]ij . (2.10.5) and indeed, the coefficient is duly projected out of M. Here is more complicated example where M is now a rank-3 tensor, and where we use a perverse mixed basis, M = Σab [M(e,u',e")]abc ea u'b e"c M ϵ VWX (2.8.9) The projector is (ei u'j e"k) and we find (ei u'j e"k) M = (ei u'j e"k) Σab [M(e,u',e")]abc ea u'b e"c = Σab [M(e,u',e")]abc (ei u'j e"k) (ea u'b e"c) = Σab [M(e,u',e")]abc (ei ea)(u'j u'b)(e"k e"c) // theorem (2.9.16) = Σab [M(e,u',e")]abc δia δjb δkc // each pair is dual as in (2.3.2) = Σab [M(e,u',e")]ijk . (2.8.10)