Linear Algebra and Geometry

CHAPTER VI. MULTILINEAR FORMS

CHAPTERV1 MULTILINEAR FORMS Linear transformations studied in Chapter I1 are, by definition, vector-valued functions of one vector variable satisfying a certain algebraic requirement called linearity. When we try to impose similar conditions on vector-valued functions of two (or more) vector variables, two different points of view are open to us. To be more precise, let us consider a mapping @:฀ X X฀ Y +฀ Z฀where X, Y and Z฀are all linear spaces over the same A. Now the domain X X฀ Y can be either (i) regarded as the cartesian product of linear spaces and thus as a linear space in its own right or (ii) taken just as the cartesian product of the underlying sets of the linear spaces. If we take the first point of view then we are treating a pair of vectors XEXand~ E Y as one vector (X,y) of the linear space X X฀ Y; therefore we are treating, 4: X X฀ Y +฀ Z฀essentially as a mapping of one linear space into another and in this case linearity is the natural requirement. As a linear transformation @฀ can then be studied with the aid of the canonical injections and projections of the product space X X฀ Y as well as other results of Chapter 11. If we take the second point of view and if at the same time we take into consideration the algebraic structures of X, Y and Z฀separately, then it is reasonable to impose bilinearity on 4, i.e. to require 4 to be linear in each of its two arguments. Such a.mapping is called a bilinear mapping. We have, in fact, encountered one such mapping before in 56A and §8A when we studied composition of linear transformations Hom (A,B) X฀ Horn (B, C) - + ฀ Hom (A,C). The most interesting and important examples of these mappings are the bilinear forms on a linear space X, i.e. bilinear mappings X X฀ X - + ฀ A (where A is the l-dimensional arithmetical linear spaceA'). The natural generalization of bilinear mapping and of bilinear form are n-linear mapping and n-linear form which are also called multilinear mapping and multilinear form respectively. The study of multilinear mappings constitutes an extensive and important branch of mathematics called multilinear algebra. In this course we shall only touch upon general properties of multilinear mappings on a linear space (฀ 5 16) arid go into considerable detail with determinant functions ($฀17) and much later in $21 and in 524 we shall study inner product in real linear spaces and hermitian 516 GENERAL PROPERTIES OF MULTILINEAR MAPPINGS 197 product in complex linear spaces which are important types of bilinear forms. 516 General Properties of MultilinearMappings A. Bilinear mappings We begin our discussion by laying down a formal definition of bilinear mapping on a linear space. DEFINITION 16.1. Let X and Z be linear spaces over the same A. A bilinear mapping on X with values in Z is a mapping $:฀X X฀ X - + ฀ Z such that @@,X, +h2x2 , ~ ) = h l @ ( ~ l , ~ ) + h 2 @ ( ~ 2 , ~ ) @ ( ~ , h 1 ~ 1 +฀h 2 ~ 2 ) =฀h l @ ( ~ , ~ , ) + h 2 $ ( ~ , ~ 2 ) for all X , xl ,฀ X , EX and X,, X2 EA. We call $฀a bilinearform on X if its range Z is identical with the arithmetical linear space A. A bilinear mapping @฀ on X is said to be symmetric if $(xl,x2)=฀ @(X,, X , ) for all X , , x2€X and it is said to be skew-symmetric (or antisymmetric)if @ ( X , , X , ) =฀-$(x2,X , )฀for all X , , x2€X. EXAMPLE 16.2. For each pair of vectors x =฀ (A,, A,, h3) and y =฀ ( p , ,฀ p,, p, )฀of the real 3-dimensional arithmetical linear space R3, we define their exterior product (or vector product) as the vector of R3.The mapping @:฀ R3 X฀ R3 +฀ R3 defined by @ ( x y ) =฀X฀ A y is then a skew-symmetric bilinear mapping on X. Moreover the exterior product satisfiesJACOBI'Sidentity: EXAMPLES 16.3. For each pair of vectors X฀ =฀(h,,฀ X,, .฀.฀ .฀,฀An) and y =฀ ( p 1,฀ p2, .฀.฀.฀,฀ pn )฀of the real arithmetical linear space R", we define their inner product (or scalar product) as the scalar of A. The mapping @:฀ Rn X R" +฀ R defined by @(xy) =฀(xlv)is a symmetric bilinear form on Rn.For n =฀2, this is 198 VI MULTILINEAR FORMS which is a familiar formula in coordinate geometry. On h2,a bilinear form $฀is defined by for...