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Appendix A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra The key word “algebra” is derived from the name and the work of the nineth-century scientist Mohammed ibn Mˆusˆa al-Khowˆarizmi who was born in what is now Uzbekistan and worked in Bagdad at the court of Harun al-Rashid’s son.“al-jabr” appears in the title of his book Kitab al-jabr wail muqabala where he discusses symbolic methods for the solution of equations, K. Vogel: Mohammed Ibn Musa Alchwarizmi’s Algorismus; Das forested Lehrbuch zum Rechnen mit indischen Ziffern, 1461, Otto Zeller Verlagsbuchhandlung, Aalen 1963). Accordingly what is an algebra and how is it tied to the notion of a vector space,atensor space, respectively? By an algebra we mean asetS of elements and a finite set M of operations. Each operation .opera/k is a single-valued function assigning to every finite ordered sequence .x1;:::;xn/ of n D n.k/ elements of S a value S .opera/k .x1;:::;xk / D xl in . In particular for .opera/k.x1;x2/ the operation is called binary,for.opera/k.x1;x2;x3/ ternary, in general for .opera/k.x1;:::;xn/n-ary. For a given set of operation symbols .opera/1, .opera/2;:::, .opera/k we define a word.Inlinear algebra the set M has basically two elements, namely two internal relations .opera/1 worded “addition” (including inverse addition: subtraction) and .opera/2 worded “multiplication” (including inverse multiplication: division). Here the elements of the set S are vectors over the field R of real numbers as long as we refer to linear algebra.Incontrast,in multilinear algebra the elements of the set S are tensors over the field of real numbers R. Only later modules as generalizations of vectors of linear algebra are introduced in which the “scalars” are allowed to be from an arbitrary ring rather than the field R of real numbers. Let us assume that you as a potential reader are in some way familiar with the elementary notion of a threedimensional vector space X with elements called vectors x 2 R3, namely the intuitive space “ we locally live in”. Such an elementary vector space X is equipped with a metric to be referred to as threedimensional Euclidean. E. Grafarend and J. Awange, Applications of Linear and Nonlinear Models, 571 Springer Geophysics, DOI 10.1007/978-3-642-22241-2, © Springer-Verlag Berlin Heidelberg 2012 572 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra As a real threedimensional vector space we are going to give it a linear and multilinear algebraic structure. In the context of structure mathematics based upon (a) Order structure (b) Topological structure (c) Algebraic structure an algebra is constituted if at least two relations are established, namely one internal and one external. We start with multilinear algebra, in particular with the multilinearity of the tensor product before we go back to linear algebra, in particular to Clifford algebra. p A-1 Multilinear Functions and the Tensor Space Tq Let X be a finite dimensional linear space, e.g. a vector space over the field R of real numbers, in addition denote by X its dual space such that n D dim X D dim X. Complex, quaternion and octonian numbers C, H and O as well as rings will only be introduced later in the context. For p; q 2 ZC being an element of positive integer numbers we introduce Tp X X q . ; / as the p-contravariant, q-covariant tensor space or space of multilinear functions X X f W X ::: X X ::: X ! Rp dim q dim : 1 p If we assume x ;:::;x 2 X and x1;:::;xq 2 X,then § ¤ x1 ˝ :::˝ xp ˝ x ˝ :::˝ x 2 Tp.X; X/ ¦ 1 q q ¥ holds. Multilinearity is understood as linearity in each variable.“˝” identifies 1 p the tensor product, the Cartesian product of elements .x ;:::;x ; x1;:::;xq/. 9 6 Example A.1: Bilinearity of the tensor product x1 ˝ x2 For every x1; x2 2 X, x; y 2 X and r 2 R bilinearity implies .x C y/ ˝ x2 D x ˝ x2 C y ˝ x2 .internal left-linearity/ x1 ˝ .x C y/ D x1 ˝ x C x1 ˝ y .internal right-linearity/ rx ˝ x2 D r.x ˝ x2/.external left-linearity/ x1 ˝ ry D r.x1 ˝ y/.external right-linearity/: | 8 7 p A-1 Multilinear Functions and the Tensor Space Tq 573 x x T0 The generalization of bilinearity of 1 ˝ 2 2 2 to multilinearity of x1 xp x x Tp ˝ :::˝ ˝ 1 ˝ :::˝ q 2 q is obvious. Tp Definition A.1. (multilinearity of tensor space q ): 1 p For every x ;:::;x 2 X and x1;:::;xq 2 X as well as u; v 2 X , x; y 2 X and r 2 R multilinearity implies 2 p .u C v/ ˝ x ˝ :::˝ x ˝ x1 ˝ :::˝ xq 2 p 2 p D u ˝ x ˝ :::˝ x ˝ x1 ˝ :::˝ xq C v ˝ x ˝ :::˝ x ˝ x1 ˝ :::˝ xq .internal left-linearity/ 1 p x ˝ :::˝ x ˝ .x C y/ ˝ x2 ˝ :::xq 1 p 1 p D x ˝ :::˝ x ˝ x ˝ x2 ˝ :::˝ xq C x ˝ :::˝ x ˝ y ˝ x2 ˝ :::˝ xq .internal right-linearity/ 2 p ru ˝ x ˝ :::˝ x ˝ x1 ˝ :::˝ xq 2 p D r.u ˝ x ˝ :::˝ x ˝ x1 ˝ :::˝ xq/.external left-linearity/ 1 p x ˝ :::˝ x ˝ rx ˝ x2 ˝ :::˝ xq 1 p D r.x ˝ :::˝ x ˝ x ˝ x2 ˝ :::˝ xq /.external right-linearity/: A possible way to visualize the different multilinear functions which span T0 T1 T0 T2 T1 T0 Tp f 0; 0; 1; 0; 1; 2;:::; q g is to construct a hierarchical diagram or a special tree as follows. ∈ 0 {0} 0S x1 ∈ 1 x ∈ 0 0 1 1 x1⊗ x2∈ 2 x1 ⊗ x ∈ 1 x ⊗ x ∈ 0 0 1 1 1 2 2 x1 ⊗ x2 ⊗ x3∈ 3 x1 ⊗ x2 ⊗ x ∈ 2 x1 ⊗ x2 ⊗ x ∈ 2 x ⊗ x ⊗ x ∈ 0 0 ; 1 1 ; 1 1 ; 1 2 3 3 574 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra When we learnt first about the tensor product symbolised by “˝”aswellasits multilinearity we were left with the problem of developing an intuitive under- 1 2 1 standing of x ˝ x , x ˝ x1 and higher order tensor products. Perhaps it is helpful to represent ”the involved vectors” in a contravariant or in a covariant 1 1 2 3 1 2 3 basis. For instances, x D e x1 C e x2 C e x3 or x1 D e1x C e2x C e3x is 1 2 3 a left representation of a three-dimensional vector in a 3-left basis fe ; e ; e gl of contravariant type or in a 3-left basis fe1; e2; e3gl of covariant type. Think in terms of 1 x or x1 as a three-dimensional position vector with right coordinates fx1;x2;x3g or fx1;x2;x3g , respectively. Since the intuitive algebras of vectors is commutative we 1 2 3 may also represent the three-dimensional vector in a in a 3-right basis fe ; e ; e gr of contravariant type or in a 3-right basis fe1; e2; e3gr of covariant type such that 1 1 2 3 1 2 3 x D x1e C x2e C x3e or x1 D x e1 C x e2 C x e3 is a right representation of a three-dimensional position vector which coincides with its left representation thanks to commutativity. Further on, the tensor product x ˝ y enjoys the left and right representations X3 X3 1 2 3 1 2 3 i j e x1 C e x2 C e x3 ˝ e y1 C e y2 C e y3 D e ˝ e xi yj iD1 j D1 and X3 X3 1 2 3 1 2 3 i j x1e C x2e C x3e ˝ y1e C y2e C y3e D xi yj e ˝ e iD1 j D1 which coincides again since we assumed a commutative algebra of vectors. The product of coordinates .xi yj /; i; j 2f1; 2; 3g is often called the dyadic product. Please do not miss the alternative covariant representation of the tensor product x ˝ y which we introduced so far in the contravariant basis, namely X3 X3 1 2 3 1 2 3 i j e1x C e2x C e3x ˝ e1y C e2y C e3y D ei ˝ ej x y iD1 j D1 of left type and X3 X3 1 2 3 1 2 3 i j x e1 C x e2 C x e3 ˝ y e1 C y e2 C y e3 D x y ei ˝ ej iD1 j D1 of right type. In a similar way we produce 3;3;3X 3;3;3X i j k i j k x ˝ y ˝ z D e ˝ e ˝ e xi yj zk D xi yj zk e ˝ e ˝ e i;j;kD1 i;j;kD1 of contravariant type and p A-1 Multilinear Functions and the Tensor Space Tq 575 3;3;3X 3;3;3X i j k i j k x ˝ y ˝ z D ei ˝ ej ˝ ekx y z D x y z ei ˝ ej ˝ ek i;j;kD1 i;j;kD1 of covariant type. “Mixed covariant-contravariant” representations of the tensor 1 product x1 ˝ y are X3 X3 X3 X3 1 j i i j x1 ˝ y D ei ˝ e x yj D x yj ei ˝ e iD1 j D1 iD1 j D1 or X3 X3 X3 X3 1 i j j i x ˝ y1 D e ˝ ej xi y D xi y e ˝ ej : iD1 j D1 iD1 j D1 1 In addition, we have to explain the notion x 2 X ; x1 2 X: While the vector x1 is an element of the vector space X , x1 is an element of its dual space.
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