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Metric Vector Spaces: the Theory of Bilinear Forms

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Metric Vector Spaces: the Theory of Bilinear Forms Chapter 11 Metric Vector Spaces: The Theory of Bilinear Forms In this chapter, we study vector spaces over arbitrary fields that have a bilinear form defined on them. Unless otherwise mentioned, all vector spaces are assumed to be finite- dimensional. The symbol -- denotes an arbitrary field and denotes a finite field of size . Symmetric, Skew-Symmetric and Alternate Forms We begin with the basic definition. Definition Let = be a vector space over - . A mapping ºÁ »¢ = d = ¦ - is called a bilinear form if it is linear in each coordinate, that is, if º %b &Á'»~ º%Á'»b º&Á'» and º'Á % b &» ~ º'Á %» b º'Á &» A bilinear form is 1) symmetric if º%Á &» ~ º&Á %» for all %Á & = . 2)() skew-symmetric or antisymmetric if º%Á &» ~ cº&Á %» for all %Á & = . 260 Advanced Linear Algebra 3)( alternate or alternating ) if º%Á %» ~ for all %=. A bilinear form that is either symmetric, skew-symmetric, or alternate is referred to as an inner product and a pair ²=ÁºÁ»³, where = is a vector space and ºÁ » is an inner product on = , is called a metric vector space or inner product space. As usual, we will refer to = as a metric vector space when the form is understood. 4) A metric vector space = with a symmetric form is called an orthogonal geometry over - . 5) A metric vector space = with an alternate form is called a symplectic geometry over - . The term symplectic, from the Greek for “intertwined,” was introduced in 1939 by the famous mathematician Hermann Weyl in his book The Classical Groups, as a substitute for the term complex. According to the dictionary, symplectic means “relating to or being an intergrowth of two different minerals.” An example is ophicalcite, which is marble spotted with green serpentine. Example 11.1 Minkowski space 44 is the four-dimensional real orthogonal geometry s with inner product defined by º Á »~º Á »~º33 Á »~ º44 Á » ~ c º Á » ~ for £ where ÁÃÁ 4 is the standard basis for s . As is traditional, when the inner product is understood, we will use the phrase “let = be a metric vector space.” The real inner products discussed in Chapter 9 are inner products in the present sense and have the additional property of being positive definite—a notion that does not even make sense if the base field is not ordered. Thus, a real inner product space is an orthogonal geometry. On the other hand, the complex inner products of Chapter 9, being sesquilinear, are not inner products in the present sense. For this reason, we use the term metric vector space in this chapter, rather than inner product space. If := is a vector subspace of a metric vector space , then : inherits the metric structure from =:. With this structure, we refer to as a subspace of =. The concepts of being symmetric, skew-symmetric and alternate are not independent. However, their relationship depends on the characteristic of the base field - , as do many other properties of metric vector spaces. In fact, the Metric Vector Spaces: The Theory of Bilinear Forms 261 next theorem tells us that we do not need to consider skew-symmetric forms per se, since skew-symmetry is always equivalent to either symmetry or alternateness. Theorem 11.1 Let =- be a vector space over a field . 1)char If ²- ³ ~ , then alternate¬¯ symmetric skew-symmetric 2)char If ²- ³ £ , then alternate¯ skew-symmetric Also, the only form that is both alternate and symmetric is the zero form: º%Á&»~ for all %Á&=. Proof. First note that for an alternating form over any base field, ~º%b&Á%b&»~º%Á&»bº&Á%» and so º%Á &» ~ cº&Á %» which shows that the form is skew-symmetric. Thus, alternate always implies skew-symmetric. If char²- ³ ~ , then c ~ and so the definitions of symmetric and skew- symmetric are equivalent, which proves 1)char . If ²- ³ £ and the form is skew-symmetric, then for any % =, we have º%Á %» ~ cº%Á %» or º%Á %» ~ , which implies that º%Á %» ~ . Hence, the form is alternate. Finally, if the form is alternate and symmetric, then it is also skew-symmetric and so º"Á#»~cº"Á#» for all "Á#=, that is, º"Á#»~ for all "Á#=. Example 11.2 The standard inner product on =²Á³, defined by ²% ÁÃÁ% ³h²& ÁÃÁ& ³~% & bÄb% & is symmetric, but not alternate, since ²ÁÁÃÁ³h²ÁÁÃÁ³~£ The Matrix of a Bilinear Form If 8 ~²ÁÃÁ ³ is an ordered basis for a metric vector space =, then a bilinear form is completely determined by the d matrix of values 48 ~ ²Á ³ ~ ²º Á »³ This is referred to as the matrix of the form (or the matrix of = ) with respect to the ordered basis 8. Moreover, any d matrix over - is the matrix of some bilinear form on = . 262 Advanced Linear Algebra Note that if %~' then vyº Á %» 4´%µ~88 Å wz º Á %» and ! ´%µ8 48 ~ º%Á » Ä º%Á » It follows that if &~ , then vy ! ´%µ8 488 ´&µ ~ º%Á » Ä º%Á » Å ~ º%Á &» wz ! and this uniquely defines the matrix 488, that is, if ´%µ8 (´&µ ~ º%Á &» for all %Á & =, then ( ~ 48. A matrix is alternate if it is skew-symmetric and has 's on the main diagonal. Thus, we can say that a form is symmetric (skew-symmetric, alternate) if and only if the matrix 48 is symmetric (skew-symmetric, alternate). Now let us see how the matrix of a form behaves with respect to a change of basis. Let 9 ~²ÁÃÁ ³ be an ordered basis for =. Recall from Chapter 2 that the change of basis matrix 498Á, whose th column is ´µ8, satisfies ´#µ8989 ~ 4Á ´#µ Hence, ! º%Á &» ~ ´%µ8 488 ´&µ !! ~ ²´%µ998 4Á ³48989 ²4Á ´&µ ³ !! ~´%µ²4998Á 48989 4Á ³´&µ and so ! 4~49898Á 449Á8 This prompts the following definition. Definition Two matrices (Á ) C ²-³ are congruent if there exists an invertible matrix 7 for which (~7! )7 The equivalence classes under congruence are called congruence classes. Metric Vector Spaces: The Theory of Bilinear Forms 263 Thus, if two matrices represent the same bilinear form on = , they must be congruent. Conversely, if )~48 represents a bilinear form on = and (~7! )7 where 7= is invertible, then there is an ordered basis 9 for for which 7~498Á and so ! (~498Á 4898 4 Á Thus, (~49 represents the same form with respect to 9. Theorem 11.2 Let 8 ~²ÁÃÁ ³ be an ordered basis for an inner product space = , with matrix 4~²ºÁ»³8 1) The form can be recovered from the matrix by the formula ! º%Á &» ~ ´%µ8 488 ´&µ 2) If 9 ~²ÁÃÁ ³ is also an ordered basis for =, then ! 4~49898Á 449Á8 where 498Á is the change of basis matrix from 98 to . 3) Two matrices () and represent the same bilinear form on a vector space = if and only if they are congruent, in which case they represent the same set of bilinear forms on = . In view of the fact that congruent matrices have the same rank, we may define the rank of a bilinear form (or of = ) to be the rank of any matrix that represents that form. The Discriminant of a Form If () and are congruent matrices, then det²(³ ~ det ²7! )7 ³ ~ det ²7 ³ det ²)³ and so det²(³ and det ²)³ differ by a square factor. The discriminant " of a bilinear form is the set of determinants of all of the matrices that represent the form. Thus, if 8 is an ordered basis for = , then " ~-det ²4³~¸88 det ²4³£-¹ 264 Advanced Linear Algebra Quadratic Forms There is a close link between symmetric bilinear forms on = and quadratic forms on = . Definition A quadratic form on a vector space =8¢=¦- is a map with the following properties: 1) For all -Á#= , 8²#³ ~ 8²#³ 2) The map º"Á #»8 ~ 8²" b #³ c 8²"³ c 8²#³ is a ()symmetric bilinear form. Thus, every quadratic form 8= on defines a symmetric bilinear form º"Á#»8 on =². Conversely, if char -³£ and if ºÁ» is a symmetric bilinear form on =, then the function 8²%³ ~ º%Á %» is a quadratic form 88. Moreover, the bilinear form associated with is the original bilinear form: º"Á #»8 ~ 8²" b #³ c 8²"³ c 8²#³ ~ º" b #Á " b #» c º"Á "» c º#Á #» ~ º"Á #» b º#Á "» ~ º"Á #» Thus, the maps ºÁ » ¦ 8 and 8 ¦ ºÁ »8 are inverses and so there is a one-to-one correspondence between symmetric bilinear forms on = and quadratic forms on = . Put another way, knowing the quadratic form is equivalent to knowing the corresponding bilinear form. Again assuming that char²- ³ £ , if 8 ~ ²# Á à Á # ³ is an ordered basis for an orthogonal geometry == and if the matrix of the symmetric form on is 4~²³8 Á , then for %~%#' , ! 8²%³~ º%Á%»~ ´%µ8 488 ´%µ ~ Á %% Á and so 8²%³ is a homogeneous polynomial of degree 2 in the coordinates %. (The term “form” means homogeneous polynomial—hence the term quadratic form.) Metric Vector Spaces: The Theory of Bilinear Forms 265 Orthogonality As we will see, not all metric vector spaces behave as nicely as real inner product spaces and this necessitates the introduction of a new set of terminology to cover various types of behavior. (The base field - is the culprit, of course.) The most striking differences stem from the possibility that º%Á %» ~ for a nonzero vector %=. The following terminology should be familiar. Definition Let =% be a metric vector space. A vector is orthogonal to a vector &%&º%Á&»~%=, written , if . A vector is orthogonal to a subset : of =%:º%Á, written , if »~ for all ::=.
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